Thanks for the reply DanU. I checked every one, they all look perfectly fine, and they don't smell like anything. The only smell on the mobo is a very faint silicon smell near the CPU, probably from the heat dispersed when power went through it - nothing smelling like burning.
I went ahead and ripped apart my old PC which was a PCI-Express 32-bit Pentium 4 (socket 775), and starting mixing and matching pieces to find my problem. The results have left me pretty confused - here they are:
Old PC parts: 450W PSU with a 20pin ATX, Pentium 4 (socket 775) cpu
Test 1) I tried hooking the old PSU into the new PC and nothing happened at all - probably underpowered with only that 20pin ATX.
Test 2) I hooked the new PSU up to my old PC and it booted fine, and posted (no buzzing).
Test 3) My new mobo says it supports Pentium 4's, and supports LGA775 socket, so I put my Pentium 4 cpu into the new mobo. I used the new PSU, and had its 4pin and 24pin ATX's both hooked in, and tried powering it up and nothing happened. I then tried unhooking the 4pin, and only had the new PSU's 24pin ATX hooked in, and it turned on and seemed to be running fine. I then hooked in the video card and a stick of RAM, to see if it would post without the 4pin ATX plugged in, and it still turned on but it did not post.
Test 4) The same as Test3, but with my old PSU. The same effect occurred, where it would only turn on with the 4pin ATX unplugged. So then I tried to see if it would post by hooking in the RAM and video card, with the 4pin still unplugged, and it would not even turn on. I'm guessing the 20pin ATX was underpowered.
Test 5) I tested the video card in my old PC to make sure it was functional, and it posted. I also checked the 1 stick of RAM, and it posted with that as well. I just had to make sure the problem laid within the CPU/Mobo/PSU.
Test 6) With the realization about the system turning on when the 4pin was unplugged, I went ahead and put the Core2Duo cpu back in, and hooked up the new PSU and the 24pin ATX. It would not turn on - but it also did not do the thing it used to with the electric buzz sound and fan twitch. I then plugged in the 4pin ATX as well, and still nothing. Afraid I had damaged it with my Pentium4, I put the pentium4 back in and tried it out again, and it turned on with the results as the last time the P4 was in. Then it occurred to me maybe I had damaged my P4, so I tried that out with my old PC again and it posted. I then went back and tried the new PSU with my old PC, and it still posted.
Then I noticed one major thing after I finished all my tests - the rear fan on the PSU doesn't spin. I'm wondering if the buzzing noise coming from the PSU before was the fan dying? Then again, the noise only happened when it was hooked into the new PC and not my old one. But I also never checked if the fans were working properly until at the very end. Could it have died from some short that's been causing a problem the whole time? Or maybe the buzzing all along was just the fan whining as it died, and it never had enough power to boot the pc?
From my tests, I think I can conclude:
-The RAM works
-The PSU works (broken rear fan?)
-There might be something wrong with my mobo's 4pin ATX plug socket?
-The Core2Duo takes alot more power than my Pentium4

-Maybe the new PSU just isn't pumping out enough power to boot the PC with the Core2Duo E6600?
-Then again, my friend's PC was able to use this new PSU to boot his new computer which uses an E6600 as well, but with a different motherboard - a less expensive one which *might* use less power. This was about 6-7 days ago, and it didn't make any buzz noise with his PC.
I'm really, really confused at this point. The only parts I haven't been able to test in another machine are the CPU and Mobo.
Thanks again for any help!