Celerons will perform very similarly in terms of performance when compared with the standard Pentiums, if you are looking at integer calcumlations. They are excellent considering price and performance together in this respect. They fall short, however, when you look at their floating point performance, which is lacking.

The earlier celerons were very good for overclocking too (most notably some Celeron 300As, which could be overclocked to 450 MHz). I'm not sure how viable an option overclocking is with the newer pentiums.

The AMD Athlons are generally recognised as being slightly better than the Pentium IIIs, and are also a bit cheaper. If you get an Athlon get one of the new Socket A chips, they have faster level 2 cache - 256k of full speed cache I think. Personally I'd go for the Athlon, because they're better value and just as fast (if not faster). I can see a CPU war thread coming here