I currently have both phones, and am trying to decide which to keep. I'm sure it's all been said by now, but here are the pros of each for me:
S3
-Smoother UI, by far. I am using Nova launcher on both, and have the same apps installed. I get occasional lag and stutters on the RMHD, but never on the SG3 - it's smooth as silk.
-better camera
-better stock sms app (I like that you can use volume up/down to control font size in messages), plus it just looks nicer to me. On the RMHD, I had to go to handcent.
-faster to pick up 4G after being on wifi connection
-better use of screen size due to nav buttons under screen, rather than on-screen.
-great battery life
-better camera
RMHD
-better call quality/stronger signal.
-feels sturdy
-even better battery life