Horses can't vomit
It is physically impossible for horses to throw up. This mechanism is vital to survival in so many other species, including humans. The ability to get rid of whatever toxic substance is upsetting your stomach, or whatever meal is weighing you down when trying to evade a predator, can save your life. So what keeps horses from having this ability?
According to veterinarian Joe Bertone in an article in Equus, "[T]he muscles of the equine lower esophageal sphincter are much stronger than in other animals, making it nearly impossible to open that valve under backward pressure from the stomach. Also, the equine esophagus joins the stomach at a much lower angle than in many animals, so when the stomach is distended, as with gas, it presses against the valve in such a way that holds it even more tightly closed. And, located deep within the rib cage, the equine stomach cannot be readily squeezed by the abdominal muscles. Finally, horses have a weak vomiting reflex---in other words, the neural pathways that control that activity in other animals are poorly developed in horses, if they exist at all."
That's the "how" for the inability to vomit. The "why" is a little more vague. Whereas humans and nearly every other vertebrate can vomit, the horse (and most rodents and rabbits) can't. The evolutionary reason for this inability in horses isn't known for sure, though one theory is that it is so they won't vomit when running at a full gallop from a predator, since the back-and-forth motion of their body could induce vomiting.