First, the number of pins on fans. 3-pin fans and 4-pin fans are designed to operate differently BUT the connectors are designed to work either way. You can always plug a 3-pin fan into a 4-pin mobo connector, and you can always plug a 4-pin fan into a 3-pin connector. The fans will work, but the details depend on how the mobo is designed.
A 3-pin fan has 3 wires: Pin #1 (Black wire) is Ground, Pin #2 (Red wire) is +DC supply, varying from 0 to 12 V, and Pin #3 (Yellow wire) is the fan speed signal. This last one is a pulse signal (2 pulses per motor revolution) generated by the fan motor and sent back on this Yellow line to the mobo for monitoring. The mobo does NOT use the speed signal to control speed, but it DOES measure speed and show it to you. Speed of the fan is based on temperature measured with a sensor (one in the mobo for the case fans, or one inside the CPU for CPU cooling system) and is controlled by changing the +DC supply voltage to the fan.
A 4-pin fan has 4 wires (surprise!) and the first three are almost exactly the same: Pin #1 (Black wire) is Ground, Pin #2 (Yellow wire - note this change) is (constant) +12 VDC supply, Pin #3 (Green wire - note this change) is fan speed pulse signal, and Pin #4 (Blue wire) is the PWM signal. The fan motor always has a supply of +12 VDC from Pin #2, unlike the 3-pin arrangement. Inside the fan case is a small controller chip that uses the PWM signal from the 4th wire to control how much of the power from that +12 VDC supply line is actually fed through the motor, and that is how its speed is controlled.
What happens if you put a 3-pin fan on a 4-pin output connector? Well, for starters, it works, but there are options in some cases. If the output port simply acts like a standard 4-pin output, the fan motor will receive +12 VDC at all times, and always run at full speed. But some mobos allow you the change the port's operating mode in the BIOS Setup screens to make it behave as a standard 3-pin fan port, so that the +VDC line does output variable voltage and control the fan speed. Some mobos even automate this process, detecting whether the fan connected is 3-pin or 4-pin and adjusting its output mode to match.
What happens when you plug a 4-pin fan into a 3-pin mobo output port? It works. The port is putting out variable voltage on the +VDC line, and the fan just uses that to run according to the voltage available. In the absence of a proper PWM signal, the 4-pin fan just uses the voltage it gets.
Now, about the PWR_FAN mobo port. It intended use is that you should plug into it a special set of wires (with a 3-pin connector) from the PSU only. Although it has 3 wires, it actually is used only to feed the speed pulse signal from the fan INSIDE the PSU to the mobo so that it can be displayed and checked. The mobo does NOT actually control the PSU fan speed. Many PSU's do their own internal control of their fan's speed, but they do not depend on the mobo to do it. And many PSU's do NOT have this special wire set, so you normally would simply NOT use this mobo connector in those cases.
BUT apparently many mobos actually provide in the PWR_FAN port the standard Ground, +12 VDC, and Speed signal connections. Thus you CAN connect a case fan to this port and it will run. However, the ones I've heard of do not do any attempt to control fan speed on this port - they just supply a constant +12 VDC - so the fan you plug in here will always run at full speed.