There are a couple of popular things to do with Facebook. One thing is to invite friends to play your game. For a multi-player game (like Words with Friends), accessing your social network friends was the main way to match up games with others. The other invite is to spam your friends with game invitations.
However, developers abused this and Facebook has had to respond to the complaints and it’s really hard now to get peoples attention.
The other thing that’s popular for developers to do is to share game results to their own timeline, hopefully not irritating your friends enough that they unfriend you. Game developers can get some exposure this way, but I’m not sure how effective it is.
Facebook for non-games can do some good things, for instance RunKeeper, an app that tracks your runs and walks lets you share your post-exercise results to your friends. Most people don’t mind these kinds of updates.
Rob