Most people do not realize that there is a difference between these two actions. Vomiting is an active process. There is gagging, heaving, and retching as the body actively expels stomach contents. Regurgitation is passive. For whatever reason, food is swallowed from the mouth but never really goes anywhere beyond that point.
Food sits in the esophagus until it simply falls back out the mouth at some point.
In the dog, megaesophagus is the most common cause of regurgitation. The esophagus is the tube connecting the throat to the stomach. When food is perceived in the esophagus, neurologic reflex causing muscle contraction and relaxation lead to rapid transport of the food into the stomach, like an elevator going down. Other reflexes prevent breathing during this swallowing process to protect the lungs from aspiration. When these reflexes are interrupted such as by disease in the esophageal tissue or nerve disease, the esophagus loses its ability to transport food.