Zsh has knobs for shared history. You can configure all terminals to have the same history at the same time, but I have it set to add history to the .zsh_history file immediately, but each shell gets it's own history. So ctrl-r in a shell lists commands entered into that shell (or preexisting when it started), and opening a new terminal starts with the shared history from all terminals. This is really the best of both worlds. Also there's a knob to only add a command if it isn't in the history, so no scrolling back over a dozen "cd .." or whatever.

Zsh has floating point math, for instance $((1.0/2)) is 0.5 instead of an error. This comes in handy sometimes in simple scripts.

Zsh is significantly faster.

Zsh autocomplete is much more advanced. Bash may have copied the pid completion and some other easy ones, but zsh has all sorts of completion hooks, for instance "svn help ch<tab>" expands to "changelist". It also runs --help and parses out GNU style help text for completing options.

Zsh completion also has optional Windows-style cycle mode, so you can do "prefix<tab><tab>"... and cycle through the matches instead of prefix<tab>a<tab><tab>b<tab>-<tab>" like in bash. I think you can actually make bash do this but it's some obscure key binding.

...basically zsh is just a bit better at everything.

{edit: oops meant to reply to grandparent}