The exact distribution you are using should not matter. But some of them have their proper way of doing things. We have some links in our wiki that deal with distro differences:
http://www.amule.org/wiki/index.php/Main_Page -> See Instalation.
Other than that, if you are starting with Linux now, you should get a recent distribution. Bug fixes happen all the time, you won't want to struggle with old bugs, will you?

Btw, machine speed is something that really doesn't matter with Linux. One of my machines, that works as a router, is a 486 DX4 100, 16 MB RAM. It runs without stop for months until a power failure happens. And runs the latest version of Debian unstable, a nice distro for that task.
Of course, if you want to use a graphical interface, this becomes an issue, but installing newer versions of a distribution does not mean that software will run slower. On the contrary, software will be smaller and faster. Remember, this is not Windoze.

Cheers!