There have been a lot of good blog posts about social games, especially what differs social games from multiplayer games. I've been thinking about this for a while and a recent story told by a friend of mine has made me think of a simple litmus test that separates the two.
This friend of mine told me that he has a fake profile on facebook so that he can play poker without his employees knowing that he is playing poker. All was good, but when he started winning nobody knew about it. He had no leaderboard because the fake profile had no friends. He didn't know where he ranked, even if he did, more importantly his friends didn't know. This was so troublesome that he said he went back to playing in his real profile. He needed that social context, the in your face, "I am beating you" feeling. The social context was a necessary part of the enjoyment. His anonymous profile just didn't have that.
So that's the litmus test. Try a game in an anonymous profile (or imagine it). If it works for you just the same you have a multiplayer game, if it doesn't it is a social game.