For anyone who might not know, a driver has two modes: pistonic and resonant (a.k.a. breakup).
Pistonic is where the whole cone moves as a rigid piston (in the lower range). Resonant is where only part of the cone (in the higher octaves). A whizzer (if there is one) is part of the resonant movement (although it is also simultaneously moving on the piston, LOL).
If an imaginary driver is producing (let's say) a 100Hz sine wave, then the cone is moving as a rigid piston. If it's producing 10kHz sine wave, then it's far above the pistonic range - the sound is produced by just a small sub-section of the cone, mostly in the center.
In between those two points, there's a transition zone from pistonic to resonant, and it might be smooth in places, and/or very rough. It might be sweet, charming, warm, vintage-sounding or utterly grating. It's interesting that people who cut off the whizzer sometimes found that the whizzer-less driver got worse (because that whizzer was actually damping or notching out some of the treble).
As DaveC113 said, lots of drivers have both pistonic and resonant ranges, and thus a mechanical crossover. The whizzer is a specific, optional feature.