They could have evolved white fur coat if they had moved to a snowy climate, as to blend in with the snow.
They also might evolve hooves because, like horses, having just one toe reduced the weight they had to carry at the end of each leg, making it easier for them to run and maneuver.
Evolving similarly to mountain goats could produce some cool traits.
-Unlike horses, goats have hooves comprised of two split toes. (I thought two hoof-toes could be an interesting delineation between horses and unicorns) The toes can operate independently, the goat can use just one to gain purchase on extremely narrow surfaces, or splay the toes to gain more contact area. A cushy sole provides traction on sloped surfaces and can deform inwards to absorb irregularities in the terrain.
-Some mountain goats can also see movement up to a mile away.
-Being smaller and lighter could help with climbing on weaker rocks.