The Main Difference is Whether to View it as Eschatological or Not
The "Christian symbolic" or "spiritual" view believes the symbolism represents aspects of the church now (during the present time, in this age), and is a common view of amillennialists. Whereas the "apocalyptic" view still sees the vision referring to eschatological (yet future) realities for the church in symbolic form, also generally a variant amillennial view.1
While "amillennial" means literally "no millennium" (believing no 1,000 year earthly reign of Christ per Rev 20:1-6), the view does hold to a form of "millennium" in that they take the 1,000 reference as symbolic, referring to the present age (thus why Ezekiel's temple is seen spiritually representing of this age Christian experience),2 or the age to come (i.e. eternity; with Ezekiel's temple spiritually representing after death Christian experience).3
You can see this difference stated in the blog post by D. A. Carson (accessed 10-7-2014) mentioned by the OP as in the other post, but it could be overlooked. Specifically, the number 3 view he notes he states, "chapters 40–48 are straightforward symbols of what is fulfilled in the Christian church [present age]," while of the number 4 view he states, "chapters 40–48 envisage the messianic future [eschatological age]."
Some More Specific Ideas
The following information comes from a presentation I did on amillennial views of Ezekiel's temple for my PhD studies in 2012, and so the descriptive titles (classifying terminology) is my own.4
#1 Present-somatic Essential Symbolism
(A "Christian symbolic" or "spiritual" view in the present discussion here).
In this view all details of Ezekiel's temple vision are irrelevant, and only what the interpreter sees as the essential aspects of the tabernacle and temple (in its entire OT context) of value as types symbolizing the body (hence somatic) of Christ and the Church in the present age by the fact that it is the dwelling place of God. Kim Riddlebarger was a one time advocate.5
According to Riddlebarger (pages 78-80), the view states that “the New Testament taught that Christ is the new temple,” with respect both to Christ’s own physical body (which certainly is a reference to John 2:19-22, though Riddlebarger does not give such reference) and His mystical body comprising the Church (where he notes 1 Cor 12:12ff, and 2 Cor 6:16). Because of this, he states, “what use remains for a future literal temple?” and later ties John 4:14 and Christ’s being living water to Ezekiel 37 and the water flowing from the sanctuary (which is actually an image found in chapter 47 verse 12, not chapter 37). What he does not do, in any way, is try to give an exposition of Ezekiel chapters 40-43, he just loosely designates the vision to representing the future “temple” of Christ’s body.
- Reducing four whole chapters of explicit details to merely a singular point that a temple is where God dwells implies no respect for any of the details given (since they are totally superfluous), making the chapters almost insanely created for all the detail given for no reason.
- The whole interpretation depends on later Christian revelation to make sense of, and thus the people of Ezekiel's day would have no idea what Ezekiel was really referring to (i.e. no reason to make a leap from temple building being reflective of a temple body).
Conclusion: The present-somatic essential view lacks any intelligibility from the perspective of an OT saint, and Scriptural sanity. While the essential symbolism itself is not inaccurate when coupled with NT revelation, it cannot be the original meaning to the text. Rather, Christ and later NT revelation used the existing temple imagery (including that of Ezekiel) to build a new picture linking building and body.
#2 Eternal-cosmic Historical Symbolism
(An "apocalyptic" view in the present discussion here).
In this view a broader survey of real or supposed temple imagery throughout Scripture’s recorded history (of which Ezekiel’s vision is but a part), is done for the purpose of pointing to an eternal temple that fills the cosmos. This is Riddlebarger’s later view, outlined in a blog post (accessed 6-15-2012; still active as of this posting on 10-6-2014, though details for any change was not checked).
Like the earlier view it also lacks any exposition of the chapters in Ezekiel or attention to any details—even though specifically the post in one part is about trying to answer a question on the "details" of Ezekiel's vision.6
Riddlebarger's admits his later view relies heavily on G. K. Beale (another amillennialist) as outlined in his book The Temple and the Church's Mission. Riddlebarger's later view is decidedly absent of any references to Christ as the final temple (which he so highly stressed in his earlier view; see #1 above), though Beale himself does still mention that point in his concepts.7
Riddlebarger's post summarizes Beale’s view that Ezekiel’s temple is “a picture of the new earth [emphasis added] as the dwelling of God.” In this later view, Riddlebarger, while speaking of Ezekiel’s vision in comparison to John’s vision in Rev. 21-22, states, “Obviously, the visions are related to each other as type-antitype (earthly language, eschatological fulfillment [i.e. the "apocalyptic view" being discussed here]).” What apparently was not obvious to Riddlebarger before is now after reading Beale’s book (of course, we can all change views upon noticing something--but if it was so "obvious," why was it missed before?). Riddlebarger states, “As Beale points out (pp. 346-345 [sic; perhaps he meant 346-355(?) of The Temple and the Church's Mission]), the new heavens and earth are now the holy of holies, as well as the new Jerusalem, and the new Eden.”
In Beale's article,8 Eden itself is seen as a temple, and the new creation returns back to that, while “Ezekiel and Revelation are developments of the first garden-temple” (9). Beale later states, “at the climax of all history, the inaugurated indwelling presence of God completely fills the entire cosmos, which appears to have been the design of the Ezekiel 40–48 temple all along” (31). So for Beale, Ezekiel’s temple imagery is symbolic of the new creation “temple” after the second advent, but the new creation is itself consisting of only the New Jerusalem, which is the “eschatological tabernacle” (6).
- Same as problem (1) of view #1 above.
- Similar to problem (2) of view #1 above, but instead of the original audience having nothing to make a "body temple" connection, here there is nothing in Ezekiel to indicate making a connection of the imagery to the entire cosmos.
- The NT picture of the body temple actually becomes secondary to the ultimate cosmic temple.9
- The Temple is not the New Jerusalem, though Beale attempts to equate Ezekiel's temple with John's New Jerusalem (both referring to the entirety of the new cosmos). While a few details may match between them (such as the representation of square/cube shapes in measurements), there is a vast difference between a temple and a city. There are also practically no parallels in the details given for each (there are a few). In other words, by again ignoring details of the visions, a few broad similarities have been linked (and in my opinion a poor interpretation made based off that link).
Conclusion: Eternal-cosmic historical symbolism suffers from similar issues as the present-somatic essential symbolism, only it in fact introduces more issues, even while attempting to bring in a larger Scriptural basis for the view.
#3 Present-kingdom Macroscalar Symbolism
(A differing "Christian symbolic" or "spiritual" view in the present discussion here).
This view is related to #1 above, but rather than emphasizing the "body" aspect, it views Ezekiel’s temple as a picture of the Church in kingdom community during the present Gospel age, utilizing some attention to larger scale (i.e. macro scale; hence macroscalar) details, while still ignoring micro details of the text.
The view is represented by Patrick Fairbairn's commentary from the mid 19th century.10 While he does give a broad (not verse by verse) exposition of the chapters, he still tends to avoid details, recognizing mistakes of other amillennialists in “an arbitrary and fanciful kind” which they “entered into the explanations they gave particular parts” (435; see also p. 450). So he avoids fanciful interpretation of the small (micro) details, while arguably still fancifully addresses larger (macro) details, or as he puts it, “the general design” has meaning, whereas the details “have no independent or satisfactory meaning” (447).11
Fairbairn sees the vision of Ezekiel as idealistically spiritual, and representing “the great enlarging of the spiritual Jerusalem and temple, the Church under the Gospel” (437).12 It is a “grand outline of the good in store for God’s Church and people” (445). For him, Ezekiel “speaks … chiefly of Gospel times, but as one still dwelling under the veil and uttering the language of legal times” (449). He reveals through a rhetorical question aimed against literal interpretation that he believes even the first temple and its activities had no “independent value apart from the spiritual truths they symbolically expressed” (446), and later notes that even with some of the “minuteness” of detail it would make it “impossible to construct an architectural plan without taking a good deal for granted” (455).
His macro scale symbolic meaning is evident. The walls represent sacredness, the gates and guard chambers reveal that the way to God is by holiness (455). The outer court represents a separation to God’s holy service (457). The inner court and its accommodation for sacrifices pictures “the higher elevation to which the Divine kingdom was to attain” (459). The various details are given simply to indicate “a full and everyway complete reconstruction of the house of God,” that is, a non-material reconstruction. He believes Ezekiel’s extra emphasis on detailing all the outbuildings is because of “the due elevation of the Divine community” they symbolize (463). The temple chambers represent that the future service of God would “meet God’s perfect approval” (468) while the extensive outer space around the complex symbolizes “the vast enlargement that was to be given to the kingdom of God in the times of the Messiah” (470). The altar itself and the consecration of it represent “how the work of fellowship and communion with [the Lord] is to proceed on the part of the people” (474)
- Still has issues handing details (small scale ones) like views #1 and #2.
- While there is some intelligibility in some of the symbolism that Fairbairn offers (walls do indicate separation and thus could symbolize sacredness; but see (3) following), in what way could an original reader see this as being the Church of the Gospel age? None at all.
- Arbitrary symbolism—Fairbairn ends up fanciful in his macro interpretations just as he criticized others for their interpretations of micro details. He would be hard pressed to find a basis in Scripture for nearly all his symbolic linking of the major architectural elements of the temple complex to the things he has them symbolizing in the Church. Indeed, he offers no such Scriptural support. In this way, his view is worse than either of the previous two views, which at least had some NT or other OT Scriptural links to tie their views to. This is undoubtedly the inevitable hole one falls in when trying to expound any details about the vision as symbolic, and hence why some amillennialists simply avoid exposition of the passage at all.
- A misunderstanding of types—Fairbairn’s contention that the temple (any temple) itself has no “independent value apart from the spiritual truths” it symbolizes reveals a major misunderstanding of types (note: he considers the temple an “embodied representation of Divine realities”  which is a type).13 This is because the antitype pictured by the type depends on the realities of that type to establish the picture. Fairbairn here is saying there really is no reality to what the temple was doing other than being a symbol, and thus it is not really a type at all, but merely a symbol.
Two types of "Christian" or "spiritual" symbolism were noted above that place the symbols of Ezekiel's temple primarily as reflecting the present reality of the Church—either as body or kingdom community. The other view fits the "apocalyptic" symbolism by focusing on an eschatological end design of the cosmos pictured in Ezekiel's temple.
I'm sure these do not fully exhaust the symbolic ideas that have been attempted from Ezekiel's temple vision, but it should help to understand the difference between the two broad categories that take the vision as primarily symbolic.
All the views suffer from ignoring the details of the chapters (or symbolizing without support for the symbolic connection), some more so than others. All views face a challenge of how the original message of Ezekiel would have been of any intelligible value to his original audience. All do not take a grammatical-historical (literal) interpretation of Ezekiel's text, and for that reason I personally reject them (but that is my hermeneutic).
1 Theoretically these views about Ezekiel's temple could still be held by a millennialist (one that believes the 1,000 years of Rev 20:1-6 is literal), but practically few if any do. The "literal (post-exile)" view noted in the other answer could also be a millennial view, but considers the Ezekiel reference to be already historically fulfilled in the second temple. The "millennial" view is purely a millennial view, which is also an eschatological view, but taking the temple and sacrificial system to still be a literal building to perform such services during the 1,000 year period. DISCLAIMER: I am a millennial view advocate.
2 See amillennialist works such as Kim Riddlebarger, A Case for Amillennialism: Understanding the End Times (Grand Rapids: Baker Books, 2003), 31; also Oswald T. Allis, Prophecy and the Church (Philadelphia: The Presbyterian and Reformed Publishing Co., 1945), 3. In this case the 1,000 years refers symbolically to the present Christian age.
3 This view really sees no time reference at all in the 1,000 years, other than a large number representing eternity, and the time period and eschatological symbolism refers to "the blessed state of the saints in heaven" (Allis, 5; but he is speaking not of his own amillennial view, but that of Kliefoth).
4 Since I am a millennialist, I will also include some refuting points against these views because (a) I do not believe they are a correct reading of Scripture, and (b) stating some of the problems shows both some of the similarity and differences in the symbolic approaches, which helps further answer the question.
5 This appears to be Riddlebarger's earlier view given in A Case for Amillennialism, which changed to the second view discussed above. Many of these same ideas can be found in Emanuel Swedenborg's views, who birthed the Christian offshoot of Swedenborgianism. See Robert S. Fischer, Commentary on the Book of the Prophet Ezekiel: Compiled from the Theological Works of Emanuel Swedenborg (Boston: The Massachusetts New-Church Union, 1925), 284-301.
6 There, one of the specific questions he is supposedly answering is from a person (Meg) who states, “I've never heard an amil explanation of the cooking pots and rooms for slaughtering the sacrifices of the people, and the chamber for the prince & his sacrifice,” that is, the details of Ezekiel’s vision. But again Riddlebarger avoids the details, claiming, “I don't have the space to cover all of the details that Meg mentions.” Like space is at a premium on a web page! This is simply an avoiding of having to handle the details, which is exceptionally difficult (and essentially purely speculative) in all the "symbolic" views. The only allusion to any detail Riddlebarger gives is the fact that Ezekiel’s description of “temple utensils” are just to picture things in “earthly terms" (an exceptionally vague answer).
7 See G. K. Beale, “Eden, the Temple, and the Church's Mission in the New Creation,” Journal of the Evangelical Theological Society 48 n. 1 (March 2005; elec. ed. 2009):19ff. This article is Beale's own summary of the view expressed in the book Riddlebarger mentions, and Riddlebarger appears to have correctly understood the view when compared to this article.
8 Ibid. All parenthetical page numbers are references to this article.
9 (All references also from Beale's work.) Beale has a self-defeating argument on his hands. He wants to agree with other amillennialists that “Israel's temple was a symbolic shadow pointing to Christ and the Church as its end-time substance” (20), and that, “our contention is that Christ not only fulfills all that the OT temple and its prophecies represent but that he is the unpacked meaning for which the temple existed all along” (27). Yet this exalting of Christ and His Church as the temple suddenly gets displaced by Beale (and is perhaps while Riddlebarger does not even note it), who then states that Christ and the Church are “a magnified view of the beginning form of the new creational temple” (27). That is, Christ and the Church are not the final temple, and thus not really the “end-time substance,” but rather the new creation itself is that substance, “the final form of the temple” (27). Perhaps the nail in the coffin of this view would be Rev. 21:22, which explicitly states “And I saw no temple therein: for the Lord God Almighty and the Lamb are the temple of it.” How much clearer could it be that the only “physical” temple in eternity is God Himself manifesting in creation and the Lamb, Christ, God incarnate eternally as man? The new cosmos itself is not the temple.
10 Patrick Fairbairn, An Exposition of Ezekiel, 1851 (Minneapolis: Klock & Klock Christian Publishers, 1979), specifically pages 431-475 regarding chapters 40-43 of Ezekiel. Parenthetical page references are to this.
11 It is almost incredible that of Ezekiel 41:15 Fairbairn notes “the account as to details is remarkably general and obscure; and why then given at all?” (466). So he ignores most of the specific, non-obscure details of these chapters, but then wonders at why a “general” picture is given without details at this one point? What!?!?—that makes no logical sense!
12 Such an idea of “enlargement” is evident in Beale’s view as well, though for him, it is not about the Gospel being spread, but rather about the actual locality of the temple, which enlarges to the entire recreated universe.
13 All types have a basis in a real, historic person, event, or thing. They must indeed become embodied in reality before they could be legitimately considered a type for anything. So it cannot be a type for Fairbairn because the temple has not existed yet in reality (not ever become embodied, i.e. built), nor ever would be.