One thing kinda hinted at but not simply stated is that a supercharger/blower is simply a belt driven compressor - the faster you turn the input shaft, the more boost you get until the point of cavitation. The faster the the engine speed - the faster the belt - the faster the belt (pulley diameter must also be considered) - the greater the output of the compressor. Different style superchargers operate better for different applications - there are always trade offs.
Your previous experience with the turbo is called turbo-lag and is because the turbo is a compressor that is spun by the exhaust gas exiting the engine. The exhaust pipe has to be big enough to exhaust adequate gas volume when the engine is revin high but the bigger it is, the lower the velocity of the exhaust gas across the entire RPM range -which means the slower the turbo spins at a given engine RPM. Again a trade off.
Long story short and a lot of tech info foregone, the boost does not come on because you hit the pedal hard. It is already there. You just add more fuel and compressed air and get a greater bang. That being said, the slower the engine is turning, the lower the pressure output of the supercharger, so there is much less performance increase.