I vote for Puppy Linux Arcade.
http://puppylinuxnews.org/puplets/puppy-linux-arcade/?category=puplets/puppy-linux-arcade/
It's a light distro so it might actually work on the hardware you have...
I am guessing the P4 has like at least 512MB's of RAM (I wouldn't use the P3 unless it's a 1Ghz or better model and has some obscene for it's time RAM.)
Edit: If the P4 is over 2GHz and has 1GB of RAM you can expect decent emulation on it. With that distro I linked. Anything less and you start losing emulators unless you can tinker with the OS and get your hands dirty. (You said your more of a console person so more hardware may be the option you should use?)
Edit 2: Really a P3 would need to be a 1Ghz+ machine with at least 1GB of RAM (I think only a small handful of motherboards at the time could handle the Copper mine CPU and that amount of RAM, most of those required like ECC RAM but at least some of them had dual CPU sockets...) I have some 1.13Ghz P3's around here (just the chip, the machines they where in died) I would also have to hazard a guess that about this time most P3's would be dead machines because that whole bad capacitor thing... Or just out of sheer age.
Edit 3: I recommended Puppy Linux because if your a Windows user it will probably be the easiest Linux for you to get used to with out getting a non lightweight distro. Not that it won't have a learning curve, but it's a little more familiar than some other distro's.
Last Edit: I just looked into it... P3's pretty much maxed out at 512MB's (It was a lot of RAM at the time) but if you where lucky enough to have access to like 20 grand you wanted to spend on RAM you could get a few motherboards that supported RDRAM 2 of them supported up to 1GB of RAM and another supported up to 4GB's. I am partially kidding about the RAM, not sure how much 4GB's would have set you back at that point in time, but I am sure it was well outside a normal persons budget.