I listen to a pretty good variety of music. TBH, I don't really listen to much classical.
I'm more a rock/electronic guy myself. I found that the highs on the HD600 were more prominent than the HD650's, and the bass quantity was slightly less. The end result was a more balanced sound compared to the "darker" HD650. The HD600 is considerably more comfortable than the HD650 too, at least for me. It has a lot less clamping force. I also think it looks better.
I wish I could have listened to the HD650 out of the GS-1 when I had it, as I have a strong suspicion that they would have made an excellent match. The HD600 almost sounded a bit dry out of the GS-1, although I've never heard it sing with more authority.
If it helps, the first "high end" headphone I ever owned was the K701, and I sold it off quickly after I heard the HD600. Why? Because the HD600 has a mellower sound and smoother highs!
At the time, I specifically evaluated the two headphones with the soundtrack to the video game
Shadow of the Colossus, which is orchestral.
However, I don't dislike AKG headphones entirely. I actually like the K501 a lot more than the K701, although I never had the chance to compare the two side-by-side. It is very competitive with the HD600, even better in some areas. But it isn't as good of an all-rounder. It is also a monster to drive, even more so than the K701 I think.