What is the difference between Cloud Code Hosting and Backward Compatible mode?

Both modes call your Cloud Code allocator module identically. The only difference is what assignment type the game client receives after a match is found. The assignment type your client receives corresponds to what your allocator returns from Poll:

• Cloud Code Hosting — the client receives the assignment type that matches what your allocator returns. For dedicated server allocation this is IpPortAssignment via AssignmentData.IpPort(), but other types such as MatchIdAssignment and CustomAssignment are also supported depending on your use case.
• Backward Compatible (Multiplay Hosting — Deprecated) — regardless of what your allocator returns, Matchmaker wraps the result into a MultiplayAssignment before it reaches the client. Use this only if you have a shipped client you cannot update.

Your allocator code does not change between modes. The pool's Hosting type setting in the dashboard controls the behaviour. See the hosting migration documentation for setup steps.