Once backed up iPhone to iCloud, can I delete an app and redownload it with same data? Let's say I'm sick of Angry Birds, I delete it from my phone. I back up my phone daily to iCloud. 

In a few months when I want to play it again, will all my data be there (or is it different implementation every app)?

I know there might be two answers since I may either: 1) go to the App store and download it from there, or 2) Sync back the app from iTunes. What would happen in either case?
As long as you don't make a single backup of your iOS device after you deleted angry birds, you can restore from a backup and the game should have data of progress you've made till the day of that backup. If you redownload the game, you would not have any app-data, regardless of anything, unless you've jailbroken your device and manually made a backup of the app-data, and reapplied it.