I can see where you're confused. A 24-pin PSU may have some of the block removeable for compatibility on older 20-pin boards. The 4 pins that come from here are a little different from the auxillary 4-pin block that are on motherboards. They do not differ between Intel and AMD platforms, but the P4 ready part comes from the days when the P4 was new and required more juice on the +12V rail.

I'd start by taking the motherboard out of it's case and placing it on a flat, non-conductive surface. This way you can eliminate any possible shorts coming from inside the case. Have 1 stick of RAM, the CPU and heatsink, video card and the PSU installed. Make sure that all power connections on the motherboard are plugged in.
You can use a flat-headed screwdriver to short out the pins on the motherboard's front panel connectors to start up this configuration.

The reason you want to go down to this barebones configuration is to eliminate as many possible problems as you can.

As far as parts go, what exact parts do you have?
What symptoms was the system having prior to it's inability to turn on?