The Note 2 is better. Without a doubt. More features, better battery, louder call and speaker volume, super amoled + non pentile screen, multi window, s-pen.
The only advantage to using the Siii is that it's easier to manage with one hand and easier to put in the pocket.
I thought I would have issue with the size, however now that I have it, I can't imagine using a smaller phone. Dwarfs my old (and wife's current) Siii's. I did love my S3, but the Note 2 is the Siii on steroids.