Personally I don't replay finite games unless they were really freaking good, meaning I remember that awesome or even just okay first playthrough, not that 100th attempt that was finally boring enough to put me off.
As I said, I don't necessarily play games to finish them. I don't even finish half of the games I play, I just stop when I lose interest in them for one reason or another. That doesn't mean I don't take anything (good) away from them though. It's like a meal, you can finish it once you're full, before having eaten everything and still be completely satisfied. I don't stop playing games when I'm bored, I stop playing them when I've had enough of them, big difference.
The two types also have different elements by necessity, the most obvious (but definitely not the only one) being story.
Why? Demon's Souls has been mentioned already. Also, Infinity Blade, a recent iPhone game. There are other games that use this kind of endless "Groundhog Day" structure, but I don't remember any of them right now, perhaps someone else could help out.
Endless games can also struggle to give you a sense of achievement - what's really the different between 1000 and 10000 points, or being 2000th or 4000th in the world?
The difference is that you performed better, duh. How is that not a sense of achievement?
It seems to me that games that go on forever can be just as good as amazing finite games, but the average neverending game is much worse than the average finite game because in the end you get so little out of it.
PROTIP: Don't play "average" games.
I dunno, maybe I'm just being unfair, maybe I'm trying to impose my way of playing games on other people. I also feel like this is more of an "arcade games vs. home games" discussion than "endless vs finite".