The nameless man in "The Rain Horse" who is attacked by a mysterious black horse is not fully fleshed out; he is simply a man returning home after a long absence. However, the horse is not characterized at all, being simply a force of nature with no higher purpose. The best similarity between the two is their determination:

It was definitely after him. He picked up two stones about the size of goose eggs and set off towards the bottom of the wood, striding carelessly.
[...]
It was looking out across the field towards the river. Quietly, he released himself from the thorns and climbed back across the clearing towards the one side of the wood he had not yet tried. If the horse would only stay down there he could follow his first and easiest plan, up the wood and over the hilltop to the farm.
(Hughes, "The Rain Horse," Amazon.com)

The horse does not leave him alone, and he fights it off with stones. Both the horse in its unwavering attacks and the man in his refusal to give in show strength of will and determination: the man wants to get to his childhood home and will not be stopped, even by a terrifying black horse; the horse wants to attack him for reasons unknown, and is able to hide and spring from cover, stealthy, quite unlike a normal horse. The horse only gives up when the man strikes it with rocks; the man only gives up after reaching his home to find that nothing has changed in his absence, and it is as if he never left at all.

In "The Rain Horse," how does nature compare to the man?

Nature in "The Rain Horse" is simply uncaring; there is no thought to the rain or mud, the thick forest or the wild horse. Without reason, Nature cannot be said to be intentionally aggressive; instead, Nature is more of a bystander, doing nothing to help or hinder the man aside from simply existing.

In contrast, the man's character is entirely purpose and emotion; his initial decision to enter the woods as a shortcut comes from his reaction to the rain, and he finds himself personalizing Nature in his anger:

A wave of anger went over him: anger against himself for blundering into this mud-trap and anger against the land that made him feel so outcast, so old and stiff and stupid. He wanted nothing but to get away from it as quickly as possible.
(Hughes, "The Rain Horse," Amazon.com)

This anger allows him to feel personally attacked by both Nature and the titular horse, which seems to follow and attack him. In reality, the horse may only be defending its territory, and rain has no mind of its own; only his own ego allows the man to attribute purpose and reason to Nature and Animal where none is likely found.