Connecting the game to Facebook has multiple benefits. Most importantly, connecting to Facebook saves your game progress to the cloud, allowing you to pick up where you left off on another device or after reinstalling the game. We highly recommend doing this to keep your progress secure, since there is no other way to recover your progress!
Connecting to Facebook also allows you to play with your friends. You can see each others Tournament scores, and you will get a Bird Coin bonus for each friend you beat in the Tournament. You can also send and receive a daily gift from all your friends.