The differences:

  * zsh has a ton of interactive chrome built-in, and a lot of it is on by default.
      This leads to some incompatibility with bash for scripting purposes
      the contextual completion stuff is infuriating when it's overly restrictive
  
  * A pure bourne shell (like dash) by nature doesn't support bashisms
      like $() as an alternative to ``
      On almost all gnu systems /bin/sh is a link to bash 
        which checks $0 and operates in sh mode
      Unfortunately bash's sh mode is completely broken
        it allows most bash-specific syntax
        so almost a lot of Linux users's #!/bin/sh scripts really need bash

The takeaway is that you should give up and use bash, or write your own shell if you're an asshole (I have done this).