A rooted phone can be a lot more customized than just non rooted, use Titanium Backup for example to get rid off all the bloatware they put on the device.
Flash custom roms (MAKE SURE YOU ARE USING THE RIGHT ROM!! A lot of Noobs will flash a Rom meant for the International Galaxy s III and than you end up with a paperweight) I've been rooting since G1 and every android phone after that I just rooted because carriers will put so much crap on it plus I am running a good version of JellyBean (Notice I said good not perfect)
Other things? You can overclock or underclock your CPU but know that all things are at your own risk since rooting just took away your warranty.. Plus the Galaxy S III has a flash counter so if you flash something it will add 1 everytime (Unless it's an official Samsung Rom) this can be reset by using Triangle Away from the Google Play store..
hop over to
Android, Windows Phone, and Windows Mobile Development News, Information, and Howtos - XDA Developers and check into it.. read... learn but don't destroy your phone..