[sludge3000]

Generaly power buttons only make a connection for a brief moment which then gets the motherboard to wake up and boot the machine up. So if it was a dodgy connection you probably wouldn't even get the fan starting.

Have you tried holding the power button in?
Will the cable for the power button from you burnt out PC reach to the MOBO on this other PC?(if you can do a quick test it's easier than moving everything into the other case just for testing)
If it doesn't reach how badly do you want to find out because you can always try moving the entire kit(MOBO, RAM, Graphics card, HDD, PSU) into the burnt out case.

Again it sounds like a MOBO issue but you can't really rule out the PSU as both PC's have been faulty and neither of them are in a working state due to power related issues.

Try testing then let us know.