What features are in zsh and missing from bash, or vice versa?

There's already been quite a bit of activity on the topic on other Stack Exchange sites. My experience of switching from bash to zsh, as far as can remember (it was years ago²), is that I didn't miss a single thing. I gained a lot; here are what I think are the simple zsh-specific features that I use most:

• _{The zsh feature I most miss when I occasionally use bash is autocd: in zsh, executing a directory means changing to it, provided you turn on the autocd option.⁴}

• Another very useful feature is the fancy globbing. The hieroglyphscharacters are a bit hard to remember but extremely convenient (as in, it's often faster to look them up than to write the equivalent find command). A few of the simpler examples:
      foo*~*.bak = all matches for foo* except those matching *.bak
    foo*(.) = only regular files matching foo*
    foo*(/) = only directories matching foo*
    foo*(-@) = only dangling symbolic links matching foo*
    foo*(om[1,10]) = the 10 most recent files matching foo*
    foo*(Lm+1) = only files of size > 1MB
      _{dir/**/foo* = foo* in the directory dir and all its subdirectories, recursively⁴}

• For fancy renames, the zmv builtin can be handy. For example, to copy every file to file.bak: zmv -C '(*)(#q.)' '$1.bak'

• Both bash and zsh have a decent completion system that needs to be turned on explicitly (. /etc/bash_completion or autoload -U compinit; compinit). Zsh's is much more configurable and generally fancier.

If you run zsh without a .zshrc, it starts an interactive menu that lets you choose configuration options. (Some distributions may disable this; in that case, run autoload zsh-newuser-install; zsh-newuser-install.) I recommend enabling the recommended history options, turning on (“new-style”) completion, and turning on the “common shell options” except beep. Later, configure more options as you discover them.

²_{At the time programmable completion was zsh's killer feature, but bash acquired it soon after.}
⁴_{Features that bash acquired only in version 4 (so are still not available on many systems) are in smaller type.}

Also the default tab completion is better than bash... for example...

~/.e.dTAB will expand to ~/.emacs.d/ in zsh, bash will just beep.

zsh lets you edit a multi-line command (see zsh line editor), bash doesn't. If you try the same trick (Ctrl-p), bash fetches the last command.