There has been much debate on here and the Gixxer750 always seems to come out on top as the best alround balance between handling and power. Saying that, it’s only you who can decide. You need to try each bike out and feel which one feels better for you:)
The 600RR is an absolutely fantastic bike, as is the GSX-R750, but the biggest part of the decision may be made for you depending upon your size.
I’m 6 foot 1 and the GSX-R750 is an absolute fit. My wife has the 600RR and it fits her perfectly. (I reckon it will fit anyone under 5 foot 10) but I find my head is too far forward so I can’t read the clocks and my legs feel just too high. It’s still an awesome bike to ride though, even for me. It’s good right through the rev range, it’s on rails going round bends at speed and it just wants to go the whole time.
Another perhaps relevant point is that the GSX-R750 is, I believe, a fairly old design whereas the 600RR was reworked in 2007.
The test ride will tell you everything you need to know. Good luck!