Prawns are larger in size, and have larger legs with claws on three pairs. They have branching gills. Shrimp are smaller, have shorter legs and have claws only on two pairs. Their gills are lamellar, i.e. plate-like.
Prawns and shrimp are both decapod crustaceans i.e. that they have exoskeletons and 10 legs. They can be found in salt water and fresh water all over the world, typically swimming in search of food. Both shrimp and prawns tend to stay near the ocean floor. They also have similar flavors, and come in a wide range of sizes from minuscule to quite large.
In commercial farming and fisheries, the terms shrimp and prawn are often used interchangeably. But of late, the term "prawn" only signifies freshwater forms of palaemonids and "shrimp" for the marine penaeids.
In the United Kingdom, the word “prawn” is more common on menus than “shrimp”; while it’s the opposite in North America. The term “prawn” is also loosely used to describe any large shrimp, especially those that come 15 (or fewer) to the pound (such as “king prawns”, yet sometimes known as “jumbo shrimp”).
Australia and some other Commonwealth nations follow this British usage to an even greater extent, using the word “prawn” almost exclusively. When Australian comedian Paul Hogan used the phrase, “I'll slip an extra shrimp on the barbie for you” in an American television advertisement, it was intended to make what he was saying easier for his American audience to understand, and was thus a deliberate distortion of what an Australian would typically say.
In Britain very small crustaceans with a brownish shell are called shrimp, and are used to make potted shrimp. They are also used in dishes where they are not the primary ingredient.
edit Video explaining the differences
In this video, Dr. Greg Jenson from the University of Washington says there's really no difference between prawn and shrimp in the U.S. The use of the two words depends upon the region and in the U.K., there is a big difference between the two.