Hi, I saw your post yesterday but didn't have time to reply till now. I had an S3 before switching to the note 2 last year.
The note 2 is surprisingly pocketable since its very thin and retains the 16:9 aspect ratio. You get used to the size very quickly (took me about a week).
One handed use can be tricky for some, but I use it one handed regularly with not much issues. On the plus side, two handed typing is much better than on smaller phones due to the bigger size keyboard!!! (the stock keyboard even has a dedicated row of number which is excellent) .
The biggest and most noticeable differences are the screen size and battery life.
The screen is brighter, and better quality, so text is crisper and nicer to read. The larger screen makes things like web browsing, reading, YouTube, videos, gaming a much better experience. Multiwindow and note taking is great.
The spen is a great well integrated feature.
The resolution is still 720p on this larger screen, so it doesn't match up to the screen clarity of the new 1080p phones. However, individual pixels are barely/non noticeable at decent reading distances. If the s4 screen didn't really win you over, note 2 is a fine choice.
Battery life is a big leap over the S3, and makes you wanna use the phone far more often and more freely. It has really long standby time as well, loses only about 2-3% overbight. The phone also charges very quickly because of its 2amp tablet charger, ~2-3 hours, so no more overnight charging needed. If S3 gives 4 hours screen time, S4 would be 5 and note 2 is 7.
The speaker is louder than the S3 which is a plus.
Things like the camera and processing have no noticeable difference.
Touchwiz interface is smooth and lag free. Regularity of updates would be about the same as the S3, i.e. usual delays by Samsung and the carriers.
Looking back at the S3 (gave it to my dad actually

) is almost surreal, the phone feels way to small now, I don't think I could go back to something smaller, and I've found this is a view shared by many other note users.
I hope that was helpful. If you have any other more specific qns, just fire away
Sent from my GT-N7100