Paul is correct, the 3rd wire sends a simple pulse from a hall effect sensor, or the electronics themselves, in the internal fan electronics as a tach signal back to the MB in PC type applications.
copperclad is correct as well on both points. Brushed motors generate static noise, as the brushes move from one pole to another across the commutator. Also there is dust generated from the brushes themselves as they wear.
4 wire PC fans work similarly to 3 wire fans and get a duty cycle, PWM, signal on the 4th wire to tell the electronics when and for how long to supply current to the drive circuit to vary the RPMs of the fan. This was implemented to reduce the harmonic drone of a steady RPM of a fan. This way the fan is constantly slowing down and speeding up due to the PWM signal. The voltage is kept at a steady 12V, to keep the drive circuitry energized at all times.
Some 2, 3, and 4 wire fans don't respond well to being driven from a pulsed voltage source. This is due to the circuitry inside the fan as was explained in the thread copperclad mentioned. The drive circuit is not energized long enough for the current to actually get passed to the coils to drive the fan motor. These fans typically respond better to a variable voltage source hence why so many PC fan controllers are a simple variable resistor pot. To that effect, even 4 pin PWM fans can be driven off of 2 wires, albeit at full speed, unless they can handle a current less than 12V and still energize the motor coils if one wanted to control the RPMs by varying the voltage.
Mike the type of fan you mentioned is a little more of a special use case and not typically used in PC type fans.
I have a couple of Top Motor fans that I have salvaged off of cheaper PC power supplies. While I have not tried them in a PWM application, as I'm running my MM2 on 24V, they work OK when the RPM is varied by a resistor or a PC type fan controller like I mentioned above.