Your board was designed for 133 fsb chips. The P4c chips are designed to run on a 200 fsb. If you put the P4c 2.8 chip on your board, with its 14X multiplier, the chip will run at 1862mhz or about 2/3s of it's rated speed. You may be better off to try one of the 2.4 prescott chips.
At 1.86 a P4c will indeed be much better than the celeron. Since you will be tossing the board anyhow, why not try a small overclock. The chip would still be underclocked of course, so would be pretty safe.
Some PIII's are faster than the 2.0GHz Celeron. I believe the Celeron 1400 is faster than the 2.0GHz Celeron. And I know Durons have been faster for a while now.
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Well, this is all the more motivation for me to upgrade. When I first got this computer I just wanted something adequate to use the internet. But I now am on the computer full-time and it's just not worth it for me to stick with this slow processor. I'm moving up! It's either the 2.8c pentium 4 or the athlon 64 3000+ for me. I'll still try to figure this one out.