If you have a voltmeter/multimeter, you could try measuring the various voltages to see if any are missing or low/high.
You could try swapping the PSUs between the two machines. If you get lucky, this will get one working machine and tell you that the machine that doesn't power up has a bad motherboard AND the PSU that's in it now is bad. If neither power up, you're back to square one.
If you have a third, known good, PSU then you could swap it into the machines one at a time.