Both shells, by default, are fairly bareboned and basic.
Both can be configured extensively, although that requires a good deal of reading their manuals. Zsh comes with better tab-completion when configured (bash requires the third-party bash-completion program to be comparable); Zsh also has right side prompts, which can be kinda cool. Otherwise, at maximum configuration, they're nearly the same (well, some minor syntax differences in scripting).
Assuming you don't want to configure one or the other all by yourself, zsh probably beats out bash in terms of options. prezto and oh-my-zsh are old and rocksolid configuration frameworks for zsh; bash-it and oh-my-bash are newer but comparable frameworks for bash.