From my experience, not usually unfortunately. Sometimes apps will be backed up, but depends on phone and what version of Android is on it I believe. For example, I can't back up my apps from my S2 and put them on my S3, but I can back up apps from my S3 :S.
You can just reinstall the apps though from the Play Store. Works well, and if you do it on computer, I find it's faster than through the phone
Anyways, onto game data. I don't think there's a way to do it without having both your phones rooted. If both phones are rooted, you can use something like Titanium Backup Root (haven't tried it myself personally, but I hear it's good). Some games though, like the Angry Birds series, have apps that will backup game files. Search for Angry Birds Backup app on the Play Store, and you should find it. It works well, especially since you can backup to Dropbox and then download again later on your new phone. Other games though, unfortunately, do not have backup apps as far as I know
Of course, if I'm wrong about this (which I hope to be - I have lots of game saves I would love to transfer from my old S2 to my S3 without rooting them both), please, someone, let us know
