Some quick things about root are:
1) customs roms - Basicaly the devs get updates and new operating system files before the general public does due to leaks from people from within so you get the newest o/s for the phone weeks to months before non rooters.
2) when a phone is EOL & has no more support from the companies, devs can keep the phone current with their own stuf, roms, apps and so fourth
3) if an app come out thats not compatible for the phone, devs can rewrite it and get it to work (most of time)
4) gettings into the depths of the phone, gives you options for different radios (for better gps, signal, calling and so fourth)
5) kernals (which give you better battery consumption & other benfits)
6) Overclocking, is here you can take the stock speed & possibly double to triple it. For example, Tbolt is 1ghz, with overclocking it can go up to 1.7ghz & with a good kernal have minimal affect on battery.
7) remove all the bloatware (junk apps) that are put in by verizon and such
my biggest thing wired & wireless tether, to give me & other internet wherever I am & at anytime without the extra charge or a plan needed.
Hope this helps