Another answer addressed the issue of Samaritan rejection of one traveling to Jerusalem, so I will attempt to address your other question:
Why does this use "his face" instead of simply "him"? What is the difference supposed to be?
The difference is a Semitic flavor and the emphatic sense of the Semitic idiom behind it. The word πρόσωπον = face is found three times in this section, in verses 51, 52, and 53.
τὸ πρόσωπον ἐστήρισεν = he set his face [to...] = he resolutely set out [to...]
πρὸ προσώπου αὐτοῦ = before his face = ahead of him
τὸ πρόσωπον αὐτοῦ ἦν = his face was [set to...] = (with reference to v. 51) he was planning to go
These two idiomatic uses of πρόσωπον (considering verses 51 and 53 together) awkwardly reproduce common Hebrew idioms. Luke, a Gentile, is well known for his frequent use of Semitic phrases.1,2
If we compare St. Luke with the other Synoptists, we are forced to admit that...there are a whole host of peculiarly Lukan Semitisms, that is, constructions and phrases...which awkward in Greek, are normal and idiomatic in Semitic.
This despite the fact that Luke may or may not have been a competent speaker of Aramaic or Hebrew. Sparks adduces many examples; a few of the more common include:
- επαίρειν τους οφθαλμούς (or την φωνήν) = "to life his eyes/voice" = "to look up at/say to" 3
- ποιεΐν ελεος μετά = "to make mercy with" = "to show mercy toward"4
The phrase "στηρίζω τὸ πρόσωπον" (v. 51) is found in the LXX only in Jeremiah and, most frequently, Ezekiel, (10x; see, e.g., 6:2, 13:17, 14:18, etc.). There it translates (sı̂m pānı̂m) and refers to the prophet (or, twice, God himself) "setting his face" toward the object of the LORD's judgement, to prophesy against it. The same sort of resolution can be seen in other passages where the same Hebrew - sı̂m pānı̂m - means something more similar to Luke’s usage (e.g. Gen 31:21, 2 Chr 20:3, 2 Ki 12:18(17)),5 The basic sense shared by all of these is on of determination or resolution to action.
The idiom "he set his face to" simply conveys Jesus’ determination to go to Jerusalem. The choice of wording reflects Luke’s particular Semitizing style.6
1. H. F. D. Sparks. The Semitisms of St. Luke’s Gospel, The Journal of Theological Studies, 44:175 pp. 129- 138.
2. There is another in this section, the initial "καὶ ἐγένετο ἐν τῷ + infinitive" ("and it came to pass while...") is a distinctively Hebrew way to begin a narrative. (13x in Luke; 1x in Mark; none elsewhere in the New Testament; LXX passim)
3. E.g. Luke 6:20, 11:27, 16:23, 17:13; cp Gen 13:10, Ruth 1:9.14 (נשׂא את־עיניו)
4. E.g. Luke 1:72, 10:37; cp. Gen 24:12, Jdg 1:24 (...עשׂה חסד עִם)
5. The LXX uses other verbs to translate this phrase in these examples. Apparently this has led some to conclude that Luke either misunderstood the meaning of Ezekiel’s phrase or that he is incorporating material from a Greek source, presumably written by an individual translating from the Hebrew of these other, more obviously consonant examples, rather than working with the LXX. See, Howard Marshall, The Gospel of Luke: a Commentary on the Greek Text (NIGTC; Grand Rapids: Eerdmans, 1978), 405.
6. Whether Luke was consciously attempting to give the text a Biblical "flavor" or simply reflecting the patois of his Jewish friends remains to be seen. (See Sparks, note 1, for his opinion.)