git grep foo
The main advantage of git grep is that it can find the patterns in the git repository, i. e. also in others than the current version of the source. This cannot be done using the standard grep of course. Also there are a lot more features in the git grep like pattern arithmetic (things like git grep -e pattern1 --and --not \( -e pattern2 -e pattern3 \)), tree search using glob (things like git grep pattern -- '*.[ch]' to search only in .c and .h files) and some more.
Here's an example session for searching in an older revision:
$ mkdir git-test # create fresh repository
$ cd git-test/