I recently purchased Theme Hospital on GOG and wasn't sure if it was best to play on unpatched games instead, or doesn't it matter?

The GOG version is perfectly fine. You may choose any of the different versions of the game. I actually am not very knowledgeable if the different versions differ much, but I tend to think they should play out almost the same. Maybe some (bad) glitches may occur only in some versions.

I use the version that the origin launcher supplies when it was one of their on the house games from a few years back. Before then though, I did just play off my disc copy.

I use the one from gog