Welcome
Access to apps that will let you do certain things you can't while not rooted. There is a great little app called wifikill which lists all other wireless devices on the WAP you're connected to and lets you "kill" their network connection without disconnecting them
One great example is taking a bit for bit backup of your phone, called a NANDroid backup (because it backs up your NAND flash memory) - another is modifying the system how you want, including certain GUI changes you can't always achieve with alternate launchers (I really like NOVA Premium with stock Jelly Bean theme).
You can also flash custom ROM's which change the UI / System menu's and give a different set of features - but I've found Samsung's TouchWiz has a lot of good features which other roms don't have (unless they're based on Samsung's).
Or you can stay stock if you don't need root. Another nice feature about the S3 is the USB OTG, pickup a USB OTG cable on ebay/amazon for less than a quid and you can plug in additional storage/BlueTooth devices etc
Did you consider the Nexus 4 at all? £280 for a 16GB device that's way faster than the S3 similar size screen, but no SD card which is unfortunately a deal breaker for me, because 16Gb just isn't enough.