In general, you'll use Windows auth on your intranet. Basic is less tied to a Windows domain client and server, so maybe it'll be more usable on the internet. Of course, you must use SSL with Basic (MUST!).
Also, Windows Authentication isn't fully supported on most non-windows clients. Even if some aspect of Windows Authentication works (NTLM) you'll still see a login box (no automatic login which is a big pro for Windows Auth). It does work on a few other platforms that understand a windows domain like Safari on a Mac and Windows phone.
If credentials need to be delegated from the web server to another server (like a SQL Server), I've found that you may run into problems for clients using NTLM (Basic or Negotiate/Kerberos often delegates easier). I've been told if my servers were configured for constrained delegation NTLM would work, but that's not been my experience.
EDIT: I've never used digest.