Exchange 2013 Public Folders unavailable, but only over Outlook Anywhere
The important thing to remember is that in Exchange 2013 Public Folders don't really exist like they used to. They're basically mailboxes.
In this particular scenario Public Folders were accessible internally, but not via Outlook Anywhere (or Outlook RPC over HTTPS if you're old).
This problem can manifests where the email address policy that applies to the Public Folder mailbox does not assign an email address that can be configured by autodiscover.