My bet would be the power supply. How long had it been running when you turned it off?
Habitat for Humanity brought us a Compaq once with a bad power supply, and another machine just like it to rob the power supply from to replace it in the first machine. When we put in the replacement, it didn't work. They pitched a fit that it worked just fine when they left with it, and the machine was actually turned on just before they turned it off to bring it in.
We asked how long the machine had been running since the last time it was turned off, and they replied, "Ugh.. I don't know, probably 3 or 4 years.. why?"
Sometimes, when you power it down, that power supply just gives up the ghost and says, "WHEW!! I'm done!"
I'd check all the connections, then yank the PS out of the old machine, or set them side-by-side and plug the dead machine into the good power supply, and see if it boots.
If that doesn't work, sometimes a stuck platter in a hard drive will cause the system to behave this way. Unplug the hard drives and see if the BIOS will post.