Apple - 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.

I emailed Bare Bones's support about a related issue, and they sent me a link to a support page, which provides a workaround for the MAS limitation. They provide a script you can download into BBEdit's Application Support directory, which adds the feature back. No need to abandon the MAS after all!