Me personally, I try to do all my PS1 gaming on my actual PS1. Occasionally, for convenience, I'll play PS1 games on my PS2 or PS3. Like CRT said, save the wear and tear on the PS2 for PS2 games.
But if you don't care about putting "extra miles" on the console, I can't see much of a reason in normal circumstances to keep a PS1 around if you have a PS2. A PS2 is an all around better PS1.
But if you're looking to get around region lockouts and whatnot, for PS1 games specifically, a PS1 console is just easier. You can install a mod chip in either to circumvent the region check (sounds like you might already have one in your PS1) but it's a real pain in the ass with the PS2. The PS2 mod chip is a 20-something wire install, depending on your soldering skills it's either a nightmare or at the very least a real pain in the ass. There's swap tricks and whatnot, but I never like using such methods. There's soft mods for the PS2, but as far as I know, there's (currently) no way of loading burned or different region PS1 games directly. And by that I mean, without having to do some sort of swap trick. For the PS1, if you have an older model with a parallel port, it can be as easy as plugging in a GameShark.
The PS2 does have a short list of incompatible PS1 games, which varies by revision.
https://en.wikipedia.org/wiki/List_of_P ... yStation_2