One of the computers in my house is having a problem where after you shut it off then go to turn it back on it won't turn on unless you unplug the main power connector to the motherboard then plug it back in then it fires up just fine.
Probably the PSU. Try replacing it with something that could take it's place. If it runs, replace the other PSU. Most power issues like that are normally right at the source of the PSU. Please let us know what happens.
It was a barebone kit that came with a crappy psu which was fine for a few months until we got a 9800gtx+ then got the new psu so it would have enough power and the PCIe connectors. It worked fine for about a month then the we had the problem. Now the computer just sits there with the side panel off so we can turn it on.