I'll admit upfront that bash may well be able to do the following, but what sold me on ZSH is that by installing ZSH and then oh-my-zsh, I had a shell that was doing things that no shell has ever done for me before:

* Fuzzy completion of paths: I now find myself typing things like: scr/mus_ser<tab> and having that expand to script/musicbrainz_server.pl. It feels very natural!

* Interactive tab completion: I now usually queue music with: nyxmms2 add /media/music/<tab><tab><tab> and I can now select an artist using my arrow keys from all the tab completions. This of course works anywhere that you can use completion (ie, filename completion).

* Corrective typing: If I make a mistake typing something, ZSH notices and asks me if I want to correct this: "mv moneyz.csv ~/backups" might say "Did you mean moneys.csv" (assuming moneyz.csv doesn't exist).

* Completion seems to work in so many more places. For example, I can use tab completion with scp and ssh and it completes paths on the remote machine. This is seriously useful.

Like I said, you might very well be able to do all this with bash and other shells. But I don't have to worry about setting it up, because it's all just done for me now.