If I remember correctly, horses make less in-game time pass when you move around. This means:
* Food is consumed more slowly.
* Enemies will not move as fast (relative to you); this makes it easier to outrun enemies that are chasing you.
I do not know if they affects enemy spawns.
Of note, in the NES version (which is *very* different from computer versions), you can't buy horses; if you go to the stable in Paws, I believe the person says something along the lines of "all the horses are dead" (don't remember if there was an "eaten by monsters" excuse here or if that's just Bard's Tale 1 that I am thinking of). Then again, the NES version doesn't have food, and enemies are not visible on the world map (but combat can start even if you don't move).