In General:
A large chunk of the reason is the profile of the cam shaft, the rotating lobe that opens and closes the valves that bring in air/fuel and expel exhaust gasses. The R6 is essentially a street legal track bike and when you are racing the engine is always at the top of the rev range. To accomplish this the cams are made to deliver the most power at the top of the rev range. Unfortunately to get most of the power up at the top of the rev range power in the middle and lower parts of the rev range must be sacrificed. This makes for very poor street riding characteristics because most of us don't tool around town at a constant 12,000 RPM. Having the engine tuned for mid range like the FZ6 means you have to shift less often and generally makes the bike better for "normal" riding.
Technically:
The cams on the R6 have more "overlap" meaning that at the end of the exhaust stroke both the exhaust valve and the intake valve are open at the same time. The fast moving exiting exhaust gasses create a vacuum in the cylinder which helps draw in the next charge of air and fuel. More fuel and air equals more power. Unfortunately this feature is counter productive at lower RPM's and saps power from a slower moving engine, which is why the FZ6 has power at 5000 RPMs then the R6. Well designed automobile engines have multiple cam profiles that can change as the engine RPM rises and falls (variable valve timing). I don't know of any motorcycles that have this design feature (and no, the Honda Interceptor's V-Tec does not work this way like it does in their V-Tec cars). Obviously the engine electronics, intake track, and head geometry are tuned as well, but only to take advantage of a different cam profile. So no, the FZ6 will never have as much absolute power as the R6 but it has more usable power (in my opinion).