This one always gets askance looks. "Well, see, horses can't vomit."
Horses have no gag reflex and a much stronger "valve" in the esophagus to keep food in the stomach. The angle of the connection between the esophagus and stomach also makes it much harder for food to come back up. A horse's digestive system is strictly a one way street - they don't burp the way we do, either.
If food is coming back through the horse's mouth or nostrils, then it usually means the horse has something stuck in its throat...or neurological damage. Both are very serious, so if a horse throws up, it means something is badly wrong with it. Stomach contents may also leak through the mouth or nose of a dead horse. In fact, the last time I saw a horse with food in its nostrils it was dead within a week.
Omnivores and carnivores vomit to get rid of poisons. Horses can't do that - but are much less prone to bacterial infection, as they don't eat meat.