Have the EVO and at the moment I'm being torn for what I think ate legit reasons.
I don't game so that's not a concern...I actually use my phone for communication purposes

...probably 40 calls a day and numerous texts and emails for work.
My thoughts are as follows:
Radios.....the EVO isn't doing any show stopping here. WIFI on mine disconnects (yup...tweaked even) and 3G disconnects at times where my wifes old Galaxy keyboard contraption is still going strong. I quite honestly never had an issue with the radios on my SII. GPS sucked but two real time reviews I have seen on the SIII shows GPS to work great...and I have a nice Garmin anyway.
Call Quality....I wear hearing aids so call quality is really important. While the EVO seems fine on the earpiece, the rear speaker although very clear is a bit subdued. I have read two reviews now where they note that both the earpiece and rear speaker on the SIII is the best they have ever experienced. Granted, this is subjective.
Camera....what HTC gave us doesn't live up to the hype. I am a widely published photographer (in my dwindling spare time) and while I never expect to use a cell phone camera for anything serious, I do use it a lot for work pictures. The EVO camera in my trained eyes is not as good as the SII camera was. It is sharper than the SIII images but overall not as good.....again subjective, but given that I shoot 100-2500 images a week of wildlife and landscapes, I have a pretty good idea of what's what photo wise.
Screen....I honestly think this will be a tossup and either will be excellent once you are accustomed to them. While I like the more natural subdued colors of the EVO, you can turn the SIII gaudiness down. I spend a lot of time in airports and reading email / web stuff on the phone and having a more contrasty screen for that alone may make the SIII better on my eyes.
Ergonomics.....I can't remotely comment on this till I have an SIII in my palm to try it out.
Menu button.....I do miss it and hate the lost screen real estate in most apps currently on the EVO.
Removable battery....HUGE. Having gone through two EVO's early on which locked up and wouldn't reboot and with no way to pull the battery...I would have loved to have one I could have yanked out to see if it helped.
Honestly...I think the desicion will come down to whatever little nuances work best for each individual user and neither one will blow the other away. The EVO will be as they have on their splash screen "Quietly Brilliant" and the SIII will be in your face fast gimmicky gaudiness....but if there are enough pluses for one vs the other for my own personal use, that's the way I will go.