Hi ddw and welcome to BleepingComputer.
This could be one of a couple of things, if you are comfortable with opening the case you can use a Volt/Ohm meter to see if you switch is working properly. The switch is a normally open momentary switch which is like your door bell, when you push it in closes the contacts and completes the circuit. Test this switch you want to have the computer unplugged from the outlet. The easiest way to test this may be to trace wires from the switch back to the motherboard and disconnect them from there and test the switch from those connectors, make a notation of the location of the pins that the connectors are on. Set the meter to the Ohm scale, place the leads together and observe how the meter responds, it should zero out. Now place one lead in each connector and press the on button, the meter should respond just like it did when you shorted the leads, if it doesn't then the switch itself if bad. If the switch is good then this could be either the PSU or the motherboard. You can test the PSU with the Volts setting, once again make sure the computer is unplugged. Disconnect the 20/4 pin connector from the motherboard and place a wire jumper between the green wire and any black wire, place the black lead in any black wire socket of the connector and the red connector in the yellow socket, plug the computer back in and watch to see if the PSU start as soon as it is plugged in. This
link will take you to a site that will show the different color and the proper voltages.
Let us know what you find.