I don't believe hg has an equivalent of the git index, although there are other ways to get similar functionality if you want it. The repository formats are different, of course, but it's easy enough to convert from one to the other. Git branches are mutable pointers that don't get recorded in the commit history, while in hg each commit records the branch it was committed to. hg has "bookmarks" that work like git's branches; I don't think git has any equivalent of hg-style branches.

For the most part, you can do anything with either one that you could do with the other, and the practical differences are mostly just in the UI, as well as a cultural component (e.g. hg more strongly discouraging history rewriting is part UI, part culture). Personally, I prefer the git UI because it has a more direct mapping onto manipulating a DAG of commits, but obviously not everyone feels that way, and I don't expect them to. When I first started with DVCS, I started with hg since it seemed easier to learn.

(Although really, my DVCS UI of choice is Magit, which I use for 99.9% of git stuff, to the point where I almost never need to actually run git myself on the command line.)