I am not a big fan of the R6. I had the 08 R6 with all the goodies and never really got to where I felt comfortable on the bike. I have an 09 GSX-R750 now, and absolutely love it. Like Meat said above the R6 is a razor, so if you put any input into the bike it will move. That is great if you are very precise through the corner, bad if you are kind of sloppy. The GSX-R600/750 isn't as precise but a lot more stable. The stability allows me to feel more comfortable at speed and gives me the confidence to push my corner speeds faster. I can muscle the GSX-R750 into the turn and get in on the right line. The issue I had with the R6 is that I am not always completely set for the turn and my movements would move the bike around which took away from my confidence and ability to push faster.
Obviously in the right hands either bike can win, but more people prefer the R6 due to its handling and top-end speed. But it isn't the bike for everyone. So do the demo ride and see what you think.