BBEdit - The authenticated save for this file failed (application error code: 20004)
Yes, the App Store version is not able to do authenticated saves. From the Bare Bones Mac App Store FAQ


  Are there any differences between the Mac App Store versions of your
  software and the versions available directly from your web site?
  [...]
  
  In BBEdit and TextWrangler, authenticated saves (the ability to save
  changes to files that you do not own) and the command-line tools are
  not available in the Mac App Store versions, in order to comply with
  Apple’s submission guidelines.
  [...]
  
  Authenticated saves will not be possible in versions of BBEdit or
  TextWrangler obtained from the Mac App Store. If you desire this
  capability, please purchase BBEdit directly from us or download
  TextWrangler directly from us.
  
  If you have already purchased BBEdit from the Mac App store and need
  support for authenticated saves, please contact our customer service
  department for assistance. We will require proof of purchase from the
  Mac App Store in order to assist you; if you include that information
  when you write us, doing so will speed the process.


Update: As of version 10.1.1 there is a script that BareBones has provided to allow users of the Mac App Store version to enable this feature. More information is available on this Auth Saves Support Page.