You can use Helium (on the app store) by Clockworkmod to back up the app data on unrooted phones (you need to plug into computer first and download a Helium app on the computer). It can backup app data on rooted phones without any issues. Note that saving the backups to external SD card and then transferring the SD card will not work - I don't think Helium has been updated to work with external SDs in KitKat yet. Therefore, use the cloud save or simply sync between your two devices.
I have run into some issues with game saves - for example, restoring a Plants vs Zombie save ended up causing a force close on the program (while PvZ 2 works fine). A couple other games refuse to back up (oddly enough, I never ran into this issue while my GS3 was on Jelly Bean 4.2, only after I updated to 4.3 and went to backup in prep for a move to the GS5). I also generally have bad luck restoring data to apps that have associated "accounts" - i.e. facebook and linkedin, so I take care to avoid restoring those and just login with my new information.
You can also use ADB on your computer to do a manual backup (Helium actually does this technically, it just spoofs your phone into thinking its still plugged in). Apparently this backup solution is a built in part of Android, just never utilized. Its not easy to use, and cannot selectively backup and restore apps unless you know the specific "com.android.xyz" app name. Google "ADB backup android" and you should find some tutorials - there is one on the Nexus forums here on XDA that walks through some of the flags and settings. Be careful to not backup system apps if you do that, you are asking for trouble.
Personally it bugs the crap out of me that there is no app data migration built into Android - yes contacts, calendar, etc can all be stored in the cloud, but why not app data? For all its rhetoric about the cloud and keeping information synced up, its a glaring oversight on Google's part. I think android theoretically restores app data on certain apps (if developers code it the right way) when the initial restore is performed after initially setting up a device, but I have yet to see that work correctly. Plus that initial restore process is full of holes - lots of apps don't re-download. You would think that, in order to sell more phones, at the very least manufacturers would want to make backup/restore as easy as possible. Samsung does have Smart Switch, but it doesn't backup app data and really only restores those items that are likely in the cloud or on SD cards anyway (pictures and contacts).
Even early 2000s blackberries had backup and restore functionality, as did old palm pilots from the 90s.
