Can anyone give me a little information on how a different launcher changes the experience from touchwiz?
Do you need to root your phone to change the launcher?
Nope. No rooting necessary. Different launchers basically just let you make changes to the user interface that TouchWiz won't allow. For instance, I use Nova Prime because I like to have multiple and scrollable dock pages. I have three dock pages that contain folders of apps. I use the docks instead of regular home pages for the most common apps I use. Then I can populate the home screens with widgets. But really, the possibilities are endless. I think most of the popular launchers have YouTube videos done either by the developers or people conducting reviews. Check those out for some ideas of what they can do.
I'd suggest looking into a few launchers and checking out the ones you think seem the nicest. I'd recommend Nova. There is a free version you can test to see if you like it. Other popular ones are Apex (very similar to Nova), GoLauncher (be careful, they're a Chinese company and there have been grumblings about how they mine data...it's a good launcher, though, with a TON of themes available), ADW, and LauncherPro (not really in development any more). Zeam is another one, but it's really simple. Oh, and Action Launcher seems to be a really popular new one (good developer).
No one can tell you what the "best" launcher is because your criteria for what makes a launcher great won't be the same as everyone else. Just try a few and see what you like.
Good luck!