The alligator is a subspecies of the crocodile, part of the family Crocodylidae.
The alligator’s snout is rounded; the crocodile’s comes to a point.
Both prefer shallow, swampy water, but the crocodile is generally more aggressive than the alligator.
Incidentally, a crocodile, not an alligator, appears on the Izod Lacoste polo shirt.