Grass cut for hay is different in that it lays and "cures" or drys for days.
Its cut and swathed/raked or sort of rolled into windrows so the air can circulate and dry it. When its dried properly its baled.
Grass that is cut from a mower is left more compact so it pretty much starts to compress and ferment immediately.
Think of the difference in length of the plant cuttings. Grass cut for hay is usually long and easily "fluffed" for drying.
Grass clippings from a mower are short and dense. Its going to just lay there.
I feed some of my grass clippings to my chickens. They eat rotten tomatoes and old lettuce too. I'd never feed that sort of thing to my horses.
ETA I think if you really wanted to give your horse grass clippings, you could immediately
after its cut rake and evenly spread it onto a concrete driveway so it could dry properly you could probably feed it to your horse....but who is going to do that?