What do the different horses do?

The Dark Horse will sometimes allow you to bypass an encounter found while moving in the world map. You get an option to flee with the text "Dark Horse" next to it, like this:

I have not seen any screens that claim to be caused by the Crazy Horse, but it seems like the Crazy Horse discovers new locations faster than the other ones.

As mentioned in other answers, the Pale Horse will trigger an event where you get one of the Nex Mex necromancy skillbooks. It's unclear if this means that the Pale Horse is required to get all the necromancy skills or that the Pale Horse helps you get those skills earlier than they would normally be discovered.

The Pale Horse will sometimes revive you from fainting after you lose a battle, allowing you to keep fighting even after you passed out from being angry. It will say "{Horse-Name} woke you up by whispering terrifying secrets in your ear."