Dolphin and porpoise both belong to Order Cetacea of the mammals but they are of different family. Dolphin's is Delphinidae while porpoise's is Phocoenidae. Thus they are completely different though "porpoise" has been used to refer to small dolphin, especially fishermen and sailors.
Both dolphin and porpoise are cetaceans, which mean they are mammals most fully adapted to aquatic life. Their bodies are spindled-shape which is the most ideal for swimming, their forelimbs are modified into flippers, they have blubber (a thick layer of fat) to keep their body heat in cold water and they don't have fur to reduce resistance while swimming. Their ear bones fill the cavity in their ears thus they can't hear in the air. The sound is transmitted through their jaws and skulls instead. They have poor vision but their excellent hearing compensates that. Both dolphin and porpoise perform echolocation to move around and feed. Furthermore, they all have to come to the surface to exhale carbon dioxide and inhale oxygen. Lastly, as they are mammals, they bear their young alive and suckle them with milk.
Dolphin and porpoise live in small groups and are believed to form strong bond between individuals. They are both carnivorous and hunt in shallower waters like the continental shelves.
Porpoise is smaller but stouter than dolphin. It has small, rounded head and blunt jaws instead of beak. Its teeth are spade-shaped, whereas dolphin has conical teeth. In addition, a porpoise's dorsal fin is generally triangular, rather than curved like that of dolphin.
The diets of dolphin and porpoise are generally different. Dolphin feeds mainly on fish and squid while porpoise hunts fish, squid and crustaceans.
Porpoise tends to be less acrobatic and more wary than dolphin. Dolphins will also kill porpoises for reasons which are not fully understood, as porpoises generally do not share the same fish diet as dolphins and are therefore not competitors for food supplies.
Porpoises are rarely held in captivity in zoos or aquariums, as they are generally not as capable of adapting to tank life nor as easily trained as dolphins.