Generic PSU are rated more towards spike, while quality PSU (though a competent reviewer rates as Ok/good/whatever) are rated as steady. What is the duration of the spike and what is being used to get a reading? If it's very short duration, like just a blimp on a kill-a-watt meter I'd ignore it and look more at repetitive peaks at full load.
However, something doesn't quite make sense unless you have this system continually crunching numbers for folding at home with a GPU intensive screen saver running. I mean that with a system averaging 600W, you must have quite a bit of power hungry parts inside, parts that should make peak power above 720W with a 600W average unless your system is always doing something that keeps it using more power than it would otherwise. So, either you're folding@home/video-encoding/etc in the background or I wouldn't trust those wattage figures.
Then there's the black art of derating. You don't want to run a PSU at full load to get better lifespan out of it and hopefully decrease fan noise, dust buildup. Due to the way PSU wattages are rounded off in 50W or 100W increments, I'd ballpark your system needing 800W to 1KW PSU... but it depends on an assumption or uncertainty about the wattage readings as mentioned above, and if your system is really averaging 600W I'd go for 1KW at least, long periods of higher load take their toll on PSU lifespan the closer you are to rated output power (all else being equal).