That could be any of a number of problems. Maybe he battery was sitting in a warehouse for months before it was shipped, and you didn't condition it so (despite what some people in the industry have been claiming), you lessened the capacity. Maybe you've allowed it to discharge too close to zero too many times. (Best life is charging at the 50% discharged point.)
Maybe you have your screen turn-off time too long. Maybe you have too many apps running in the background.
Or maybe an S3 battery just can't hold enough charge for the use you need. (The S3 is rated at up to 21 hours of talk time, but that's under lb conditions with a brand new properly conditioned battery. And they probably get that like I get 75mpg in my car - going down a steep hill with a hurricane wind behind me.
1) I see TouchWiz on my phone maybe once a month - someone here asks a question that requires that I run it to see what's causing the problem. Where is this menu item or something. The rest of the time, TouchWiz s just taking up storge.
2) An app being in storage (IOW having been installed) doesn't use battery. It only uses it when it's running. And you can disable an app in Aoolication manager if you don't even want to see its icon.
Rooting to gain storage space by uninstalling system apps is one reason for rooting. It's also a way to brick the phone if you uninstall the wrong system app. (And system apps don't take up that much space.) But not for gaining battery time. Carry a spare battery or two - they're cheap. (About $25 for 2 S3 bayyeries and a charger - look on Amazon.) Five seconds to swap batteries and pop the dead one into the charger.
Comparing battery usage between an Android phone and an iPhone is like comparing apples and cars - Android phones get less battery life than iPhones, because there's more computer running in an Android phone. (iOS is the kernel, ROM and a lot of apps, all by itself. Android is a Linux vomputer running a Java Virtual Mchine [that's 2 computers right there] and almost everything is an app.) If battery life is your main concern, your real solution is to go back to an iPhone.