I don't play games on my mobile, but if I did I would go for Note 2. As I have used it for more than 7 months, I can concur that it was an amazing device. It was smooth, fast and didn't get as hot as my S4. I sold it and bought S4 for only one reason, size. It was way too big for me. I love the overall size of S4. But it stutters more than Note 2, especially I miss the fantastic stock browser of Note 2, the stock browser on S4 is next to horrible. God knows what Samsung have done!
Anyway, I am happy that I bought S4 and sold Note 2, I don't regret my decision a tiny bit, but if someone is happy or comfortable with the size of Note 2 then I see no reason to buy S4 over it.