The easiest debug step to take would be to try a known good PSU.
I'm assuming that when you say it starts after unplugging/plugging back in, that you plug the PSU back into the AC mains, and the computer does NOT start up then...you still have to press the button to turn it on?
If that's the case, it sounds like the PSU power-control circuit in the PSU is damaged, and when you unplug/plug the PSU you're basically giving the PSU a hard-reset.
If that is not the case, and the PC turns on the moment you plug the PSU into the AC mains, but your power button still turns the PC off, then I would suspect the power control circuit on the mobo isn't generating a properly controlled edge-triggered power signal to the PSU power control pin on the main power connector. This could mean that the PSU isn't maintaining the minimum standby voltage required for the mobo to be able to do so. Testing with another PSU would tell you if this is the case.
edit to add-
easiest debug step try a known good PSU
So check your BIOS/chipset settings - do you recall changing them right before this issue started?
I see the guy above also mentioned standby power as well (but didn't think of the solution of not enabling USB; etc when in standby / low pwer mode). Look around in your BIOS/chipset settings.