I always preferred putty's terminal window, not sure why, even though git bash's can be customized as well.. can you save sessions in git bash?
the problem is that putty's terminal window (as far as I know) doesn't work in local, so if you are doing local terminal-based work (vagrant git etc) and then you need to ssh, you can't do everything with putty's terminal, but you can do all of the above with git bash.