Ya, I love touchwiz I have the galaxy sii skyrocket on AT&T but I hate the cartoonish icons on it. I am looking for a lot of customization .
Touchwiz on the S3 is watered down alot from what you got on the S2. (I have the Sprint variants.)
I also own the Galaxy Nexus (stock unlocked GSM version from Google's Play store running on a prepaid T-Mobile account).
For me, the S3 is my more user friendly phone. I find Touchwiz's enhancements make the phone work better. It especially shows up when you compare the camera apps. Touchwiz' version has more features and filters. I love the ability to take a picture by voice. Helps when I'm holding the phone in an awkward position.
I also got to use the S Beam functionality with my brother's phone over the weekend. Works so well for trading photos and videos. My GNex has Android Beam but that only seems able to handle small data transfers like contacts and web pages. Every time I try to beam a photo it fails due to the large file size.
The Nexus phones will give you Android updates quicker but there's a huge caveat to that. If you are buying the phone through a carrier, then you'll still be stuck waiting for them to test and clear their bloatware versions. (Just look at how Verizon's GNex customers got screwed over the Jellybean update.) The only way to guarantee you'll get the latest OS push is if you buy an unlocked GSM phone (and use it on T-Mobile or AT&T in the USA).
My suggestion is to wait until November to decide. By then Samsung will probably be pushing Jellybean out to all the US carriers (Sprint & T-Mo will probably get it before Verizon. Like I said, bloatware is the culprit and Verizon is "King of Bloatware".). Then compare the handsets in person for their hardware feel and watch a few review videos online. Research it to death. You'll want a phone you can live with for 18 months at least.
The LG Nexus does not thrill me right now. As much as I like Jellybean and the Google Now features, not having a SD card slot is a deal killer for me. I like to use my phone as my mobile camera. I NEED a 32GB SD card to handle all my photo and video storage for a trip.
No I don't have a bias against LG. My first smartphone was an LG Optimus. It was a handy little phone that held up to a lot of drops and kicks. However I do prefer Samsung's bigger, sexier design and Asus' revolutionary styling. Only manufacturers I truly dislike in the mobile sphere are Motorola (built to look and feel like ugly bricks) and HTC (Sense is so comically overburdened with widgets it doesn't even feel like Android).
Good luck with your phone choice!