Alligators (Alligator mississippiensis, like in Florida) DO NOT live in Costa Rica. The Spectacled Caiman (Caiman crocodilus) DOES live in Costa Rica. The American Crocodile (Crocodylus acutus) DOES live in Costa Rica. The easiest way to tell the difference between an alligator and a crocodile is that alligators have a U-shaped jaw (flat snout) and crocodiles have a V-shaped jaw (pointed snout) with teeth that protrude when their mouth is closed as seen here.