What are the characteristic differences between how Jewish scholarship approaches the text of the Tanakh (Old Testament) and that of Christian scholars? If their approach was the same I assume they would come to the same theological conclusions, so while they both believe the same text to be inspired there must be a difference in the methodology they use to interpret the text.
Texts that aren't dense legalese, e.g. books like the Bible which contain stories, parables, philosophies and statues, are necessarily rich with ambiguity and mystery. There is no way that a book like the Bible could unambiguously inform any philosophy/theology/weltanschauung. The
In general: philosophy informs one's reading of a text, not the other way around.
Almost every word of the Bible, a text which lives beyond a great temporal and cultural divide, is ambiguous and open to a wide range of plausible interpretations which are debated within Jewish circles and Christian circles alike.
There are so many differences within Jewish scholarship and equally many differences within Christian scholarship that's it's impossible to characterize a single religious methodology for either. Scholars from both faiths differ in:
Moreover, who counts as a Jewish scholar? What is the status of important scholars like Robert Alter, Umberto Cassuto and Jacob Milgrom? Bible scholars that aren't religious Jews, do they count? How about Bible scholars who don't believe in the inerancy of the Bible? Bible scholars who don't believe in the divine source of the Bible? Karaite scholars? Even from within the Jewish world of Orthodox Bible Scholars who believe in the Bible's divine origin, some writers like Mordechai Breuer are marginalized and ignored by large parts of the Orthodox Jewish world over various theological questions.
The assumptions that inform one's reading of a text are a type of oral tradition that live alongside any written document. Orthodox Jews are informed by an “oral-Torah” (תורה שבעל פה), much of which is codified in the Mishnah, Talmud, Midrash and other works of Rabbinic literature that emerged immediately prior and somewhat into the Middle Ages. The theological and exegetical assumptions therein inform a characteristically “Jewish” reading of the Old Testament but these exegetical assumptions are a reflection of Jewish theological tenets and not the source of Jewish theological tenets.
What are the basic theological issues that divide Jews and Christians and why? - is a whole different question.
The basic difference is Jesus Christ. That may sound trite or rude, but it needn't be. A Christian hermeneutic that is faithful to itself will base its reading of the Old Testament on the way Jesus and the Apostles used the Old Testament. This hermeneutic was rather shocking even to Jesus' disciples (i.e. Christians) even at that time (and I assume Jewish hermeneutics have continued to develop since then, though I am ignorant on that front). Of course, a Jewish reading of the Tanakh will not use this approach, the basic premise of which is: The entire Old Testament is about Jesus, who is the Messiah. The prime source of this hermeneutic (though it is well distributed throughout the New testament) is Luke 24 where Jesus himself clearly teaches it (especially verse 44). Obviously any Jewish hermeneutic must have a radically different point of departure.
This will be a partial answer, intended primarily as a supplement to this answer, since she mentioned that she is an expert in Jewish approaches but not in Christian approaches. (I can't add anything to her answer on Jewish approaches, so I won't bother trying to cover that material!)
There are a variety of Christian approaches to the TaNaKh (i.e. the Christian "Old Testament.") Here is a rough overview of the main approaches I am aware of:
"Marcionitic": The content of the TaNaKh is so foreign to some Christians that they tend to regard it as incompatible with the Christian faith. I call this approach "Marcionitic" because it unwittingly follows the tradition of the heretic Marcion -- though its adherents are not technically "Marcionites," since they do not vehemently reject the TaNaKh; These Christians just avoid the TaNaKh altogether in their reading and teaching.
Postmodern: This approach regards the TaNaKh as shifting, evolving, multifaceted gem that will "mean" something different to everyone, since we all live in a different world and have a different perspectives, presuppositions, and agendas.
Devotional: This approach regards the TaNaKh as a Divine text which God can use to speak personal, specific things to "me" which may never have been intended by the human author -- or by God, prior to "me" reading it.
Historical-Critical: This Enlightenment-rooted approach to Scripture regards human reason as the only reliable means of determining truth. Under this approach, the TaNaKh is scrutinized by the same criteria that any other ancient text might be scrutinized. The supernatural tends to be ruled out on principle, and the authenticity and reliability of the Scriptures are definitely in question. The goal is to scientifically reconstruct the history of the text and its evolution over time.
Gnostic / Cultic: This approach regards the TaNaKh as a text that can only be understood once you've discovered the secret knowledge of the "answer key." Each cult will have their own approved "answer book" which their members (an elite class of Christians to be sure) are privileged with. (Typically it has been written by the cult leader.)
Allegorical: This approach regards the TaNaKh as a Divine product, but not intended to be taken literally. For some, this is because the literal is too base / carnal to be "the word of God." For others, this is because the literal is in conflict with other things we "know."
Jewish: This approach is the Jewish approach, but is used by a significant number of Christians who are determined to preserve the Jewishness of the TaNaKh, and treat it on its own terms, with no consideration for the New Testament, since it was not around at the time that the TaNaKh was written.
Historical-Grammatical: In general, this approach treats the text as a human text -- but a human text that is of Divine origin. As such, it is interpreted by the rules of grammar, with respect for historical context, but without skepticism about its contents or authenticity.
Sensus Plenior: In general, this approach begins with Historical-Grammatical, but sees another "layer" of meaning on top of the base "layer" of the text. The term is used to describe a wide variety of approaches, and everyone seems to have a different definition for the term. (It doesn't help that the term is often used in a derogatory way, so no one wants to be associated with it!)
The only real similarity that these approaches have over and against Jewish approaches is that the people performing them are Christian. :)
The key general differences between Jewish and Christian thought concerning the Hebrew Bible (Tanakh) lie not so much in their respective methodologies of interpretation (which are very similar), but the precedent and priority each gives to the respective biblical covenants (Abrahamic, Mosaic, Davidic, and New Covenants).
For example, according to the New Testament the Mosaic Covenant is the central theme of the Jewish faith. Therefore the grounds of ones right standing before the Lord in Judaism are derived from an accurate and an appropriate application of the Law of Moses to ones life (Rom 9:30-32). In other words, righteousness before the Lord is derived from ones alignment and faithfulness to the Law of Moses. This outlook therefore gives the Mosaic Covenant necessary precedence over the Abrahamic Covenant.
On the other hand, the Christian New Testament gives precedence to the Abrahamic Covenant over and above the Mosaic Covenant (Phil 3:9). That is, the basis of righteousness is not ones response and compliance with the works of the Law (Gal 2:16), but faith in the promises of the Lord given to Abraham. These promises centered on the "Promised Seed" of Abraham, which seed was to sprout in the Promised Land. The seed included not only the plethora of his descendants but also an individual seed (Gen 22:17), which was later "fleshed out" and identified in the Davidic Covenant (Rom 4:13). Abraham therefore had received his righteousness from the Lord through his belief and trust on these initial promises, which would have its eventual result in the blessing to the nations of the earth (Gen 22:18 and Rom 4:13). By believing these promises, Abraham was justified by faith, and the sign of his faith was circumcision.
Abraham's circumcision therefore was not only the removal of the foreskin, but the removal of the hardness of the heart (Deuteronomy 10:16), which is an invisible circumcision (Deuteronomy 30:6). The Christian New Testament therefore indicates that the "authentic" Jew is not the one who is necessarily physiologically circumcised per the Mosaic Covenant, but the one whose heart is circumcised per the Abrahamic Covenant (Rom 2:28-29). The circumcision of the heart is not an overt act of compliance with the Law of Moses, but by faith on the promises made to Abraham. Therefore "salvation" in the Hebrew Bible was through faith.
Therefore the Abrahamic Covenant, which is based on the hope of promises (and therefore was unconditional), is superior to the Mosaic Covenant, which was based on overt compliance to laws (and therefore was conditional). This is the first distinction between the respective interpretations of Tanakh.
The second distinction is the Davidic Covenant, which was an extension of the Abrahamic Covenant, and therefore was unconditional. The Promised Seed of David was the individual seed first identified in Gen 22:17, because the Hebrew word for "enemies" in that verse is modified by an adjectival suffix in the masculine singular. (In the immediate context, that seed was the individual Isaac, but in the expanded context of the Davidic Covenant, that individual seed was "fleshed out" and identified as the anointed son of David.) The Christian New Testament therefore associates this Promised Seed in type with Isaac who had been offered as a sacrifice to God; and then of course with the anointed son of David, through whom the nations of the earth would be blessed (cf. the same verse in Gen 22:17-18).
Finally the Christian New Testament views the anointed Son of David as becoming a priest according to the order of Melchizedek (Ps 110:4), who at the time of Abraham was both a priest and king. So the anointed son of David is not only the sacrifice (per the typology of Isaac), but he is also the priest according to the order of Melchizedek, who was superior to Abraham (Heb 7:6-7). This appointment of the anointed son of David, who incidentally is also superior to David (compare Ps 110:1 with Mt 22:41-46), was to be this priest, and therefore his appointment changes the priesthood of the conditional Mosaic Law (He 7:12-22). The result had forced the inauguration of the New Covenant after the presentation of his blood sacrifice on the cross (as the priest cum sacrifice).......
The discussion could continue on and on, but in the interests of brevity, the central focus of Tanakh is not the conditional Mosaic Covenant, but the unconditional Abrahamic Covenant, which provides righteousness to those who believe by faith apart from the works of the Law of Moses (Gal 4:23). The Abrahamic Covenant, in turn, spawned the unconditional Davidic Covenant and the unconditional New Covenant. What ties these three unconditional covenants together is the "Promised Seed" who is at one and the same time the suffering sacrifice (à la Isaac), the holy priest (à la Melchizedek), and the majestic king (à la David) -- thus the three unconditional covenants of Tanakh merge together in the "Promised Seed" that sprouts in the "Promised Land" (Abrahamic Covenant).
These general brushstrokes of observations and relationships between the central unconditional covenants are therefore what distinguish the Christian and Jewish approaches to Tanakh according to the Christian New Testament. The Abrahamic Covenant therefore takes precedent over the Mosaic Covenant according to the Christian New Testament. These differences are not the methodology of interpretation between Jewish and Christian scholars (which are very similar), but the assumptions behind the methodology of interpretation (hermeneutics).
In summary, the Christian New Testament indicates that the circumcision of the heart is necessary to see the glory of the Nazarene, who had unified the unconditional Covenants of Tanakh (2 Cor 3:12-16). This circumcision is the removal of the hardness of the heart (2 Cor 3:14), which comes by faith and therefore results in righteousness (à la Abraham).
A Fundamental Flaw
I believe there is a fundamental flaw in your assumption behind the question itself. Namely (emphasis added):
and so you ask the question (emphasis added):
Yet at the most basic level, everyone (of sound mind) approaches the text exactly the same way, and yet comes to differing conclusions. All are using the same "how"...
This is because each is evaluating the textual assertion as true or false, and doing
This type of "analysis" is happening at least subconsciously and often nearly instantaneously when anything is read, except when one pauses to really consider what one has read (then it is conscious and could consume a lifetime of study).
So if the Basic Approach is the Same, What Differs?
There are numerous things that do differ, which are then reflected in the basic approach above. Some key ones are:
So what appear to be differing "approaches" to the text, such as listed in other answers here like this answer or this one, are all just forms of "conclusions" that have been arrived at through the basic approach above, conclusions that are considered resolved into "knowledge" about how to answer the next Bible question that comes along (though often people are simply just taught these methodologies, and so they really enter into one's scheme as a "presupposition," not ever having been thought through to be an actual "conclusion" for oneself).
The Fundamental Issues
So presupposing a Biblical worldview (you will catch my meaning of this by reading further), people do not come to the "same theological conclusions" because of (1) being a finite creature, and (2) being sinful. This is because:
So there is no "characteristic difference" between Jewish and Christian interpretation, because both follow the basic approach. If such were not basic, no communication would or could occur... which, by the way, is why I reject Amichai's statement in his answer, "There is no way that a book like the Bible could unambiguously inform any philosophy/theology/weltanschauung." What is written is unambiguous already (it is designed to communicate), it is just the limitations noted above that prevent people from seeing clearly what is being communicated.
The limitations, however, have brought the two groups using the same basic approach to differing conclusions and beliefs, such as some other answers have noted (who the Person of Christ is, what the nature and relation of the covenants are, etc.). And these conclusions and beliefs have then become "base factors affecting the way" texts are interpreted (as you note in a comment), but not just between Jewish/Christian, but within those groups, and even between any two individual, finite, sinful beings.
Didn't see a reference to this great passage anywhere, so I thought I'd tack it on for any future readers.
Spoken after Jesus's resurrection, but before his ascension, this highlights the perspective Jesus himself added to the Old Testament. A perspective that Jewish scholars don't incorporate (in more than a general way) into their studies.
Jesus said all of the Scriptures were about him. Knowing this would surely affect how I study the scriptures, and as a result which conclusions I draw from them. If you haven't already, read some of the previous answers for a more in-depth analysis on this subject.
Jewish scholars use the method of Pardes which is an acronym for Pashat, Remez, Drash, Sod. Pashat is the literal interpretation. Christians have learned much from Jewish expositors in this. Rabbinic exposition of the literal meaning is not much different than Christian.
Remez looks at hints and follows their lead. For instance, Jesus's quotes of OT scripture while on the cross would be considered mere pointers to the whole context of the quoted passage. Generally, eschatological biases permit Christians to chop up the referenced scriptures into present and future portions.
Drash means to compare and contrast similar passages. For instance, when water is parted for Moses, Joshua, Elijah and Elisha, they are seen as transparent overlays of the same story and context can be moved from one story to the next. Christians might notice similarities and differences between them, but sharing context is nearly unheard of.
Sod is the hidden meaning. Jewish interpretation of the sodim leads into Kabbalah. For Christians sod is about Jesus, e.g. in the saying "When you see the white rock, don't say Water-water", using the methods of pardes upon the New and Old Testament together, Jesus is the white rock, and "Water-water" means the word of God in heaven and on earth. The warning in the parable of the four rabbis is followed by penalties.
Christians generally note the odd usage that the NT authors put to OT scripture, and it is the source of much debate under the name of sensus plenior. Many modern theologians have rejected the idea of hidden meaning as a response to much bad allegory. However, Clement seems to have retained a knowledge of it, and more recently Macintosh came very close to seeing the hidden since his methods indicate his ability to share context in Drash. Some discussions on intertextuality come close to describing sod, such as Jeffrey Meyer's description of Mary's encounter with Jesus at the tomb as the fulfillment of prophecies in the Song of Solomon. http://web.me.com/jeffmeyers/Site_3/The_Gardener_&_his_Beloved.html