The capacitor has two purposes. First, it provides the necessary "jolt" to start the motor. Second, a smaller capacitor inside the run capacitor is is used to stabilize the movement of the motor as the phase flips with the AC current. As far as speed regulation goes, I'd use a voltage regulator (variable resistor) over a dimmer switch. Most dimmer switched augment the frequency of the AC current vs dropping the voltage.
Someone with more physics/EE background feel free to correct me, but that's how I understand that it works.