I backed up one galaxy S3 and did a restore on another. Then I try to boot the new one and I get a message "Unfortunately the process android.process.acore has stopped". After hitting OK a few times I get Unfortunately TouchWiz has stopped. Then back to the first message. I can't get past this msg. I thought that maybe I should do another Restore but I can't get to anything cause this happens as soon as I boot.
In short, never ever back up system apps and settings from one handset and restore to another, that is exactly what happens! :)
Mismatches with system app's key signature conflicts with another's ROM's key signature, and variants of settings as well may not exist on the other, all of those can lead to breakages such as what the OP is experiencing.
The only way to get that back, is to perform a factory reset to restore everything back to its original state.