For a web page that exists, but for which a user that does not have sufficient privileges, (they are not logged in or do not belong to the proper user group), what is the proper HTTP response to serve? 401? 403? Something else? What I've read on each so far isn't very clear on the difference between the two. What use cases are appropriate for each response?
According to RFC 2616 (HTTP/1.1) 403 is sent when:
The server understood the request, but is refusing to fulfill it. Authorization will not help and the request SHOULD NOT be repeated. If the request method was not HEAD and the server wishes to make public why the request has not been fulfilled, it SHOULD describe the reason for the refusal in the entity. If the server does not wish to make this information available to the client, the status code 404 (Not Found) can be used instead
In other words, if the client CAN get access to the resource by authenticating, 401 should be sent.
See the RFC:
401 Unauthorized:
If the request already included Authorization credentials, then the 401 response indicates that authorization has been refused for those credentials.
403 Forbidden:
The server understood the request, but is refusing to fulfill it.
Update
From your use case, it appears that the user is not authenticated. I would return 401.