Git is a version control system. Github is a proprietary hosting service that manages central git repositories. A git repository doesn't have to be public, but there are some great free hosts available if you do want to make your work available. Github, obviously, or Gitorious, especially if you're offended by github's ugly gender politics. Github's GUI is much nicer than any generalized Git GUI's I've seen, which is a vote for using Github, but it really doesn't matter.
So here are a few good reasons you should use git to manage teaching materials:
- Greg Wilson explains it pretty well to a Python conference. You have to listen to him talk about a few other things first, but the whole talk is interesting so go ahead.