I'd say that the one emphasizes the nearness (proximus) and the other emphasizes the sharing of a physical boundary (finitimus).
If I had to catch the difference in English, hm, I don't quite know. Next-door neighbor is what I was going to say, but it has both the sense of proximity (next) and property (door).
Otherwise, I'd say you're right. They mean about the same thing.
Edit: I seem to recall that Jerome favors proximus for neighbor in the Vulgate, though I'd be hard pressed to say he doesn't say finitimus. I'm too lazy to do it, but if you could trawl through the New Testament and compare the uses of those words against the Greek it could be instructive.