Lost game progress after reinstalling app despite app having Game Center integration
Game Center only saves games achievements; it never saves game progress. Game developers use different methods to store your progress:


iCloud Storage: Your data are stored locally on your iCloud account. Usually you just have to reinstall the game and you'll be able to continue the game without problems. E.g. Pocket Planes
Facebook Connect: The developers will connect your Facebook account and associate the game progress with your Facebook account. As long you don't delete your Facebook account, the progress is always be available once you connect the game with your Facebook account. E.g. The Sims Freeplay
Games Account: Companies like Gameloft let you save progress to your Gameloft account. You'll need to create a separate account for these type of games. EA Games and Gameloft are the common type of company that use this approach to let customers manage their games across devices.