With Linux, 386 is the lowest common denominator. Below that lacks hardware, instructions, etc that Linux now requires. There are specialized Linux kernels that will work on older hardware, such as ELKS, but you can't do a whole lot with a 200k kernel.

If you're only going to use a PIII, go ahead and get the version optimised for it. But if you might install this on other machines, get the 386 - that's the safe way to go.