AOSP = Android Open Source Project.
It's stock Android, whatever the latest version is. Currently that's a variant of Jellybean.
I think what the PP refers to is the little enhancements you lose going to stock and away from Touchwiz. I have both the Galaxy Nexus (unlocked GSM version straight from the Google Play store which means absolutely NO carrier bloatware, currently being run on a T-Mobile SIM card) and Galaxy S3 (also on Sprint like you). I find I prefer the little enhancements I get with Touchwiz. Things like the camera app, the phone/contacts app, the Quick Tools in the Notification Bar, the voice command features, etc. My GNex gives me Google Now which is very nice, but that's really the only big feature advantage. I'm eagerly awaiting Jellybean on the S3 just to see how well Samsung marries the best of Touchwiz to the best of Jellybean.
If I were you, I'd go for the Note 2 right now. (In fact, I'm kinda jonesing for it myself.) It looks exactly like the S3 but it's bigger and has more features (including Jellybean). I love the split screen feature and the S-Pen tools.
Downside is the bigger phone size. If that's a barrier to you, then go get yourself another S3. (And don't be such an idiot you trade it for a crappy iPhone again!!!)
Your other alternative is to wait a month and see what sort of Nexus phones get announced. But honestly, I don't think it's worth getting a Nexus phone from a carrier. They always hold up OS updates for their testing and bloatware. Sprint did, though not as long as idiot Verizon. Best use of a Nexus phone is an unlocked GSM version you can run on AT&T or T-Mobile with one of their SIM cards. Sprint and Verizon will always give us issues since they are CDMA carriers.
Kudos on having Sprint LTE! Sprint just started rolling it out to my area. I only get LTE in the city but my 3G speeds have improved with the network upgrades. In a year's time, we should see a much bigger area roll-out. Ignore the naysayers around here about that. There's a lot of Sprint hate around, but usually from Verizon folks who have been sucked into the vortex of "Here Now but Pay Through the Nose" economics to appreciate competition.
