Key west has more night life. If you like to drink and party all night, there are a lot of places in Key west to do just that. Key West is beautiful but the few beaches are small and often crowded. Key west is not a very big island and you can see it all in a day or less. If you rent a car and drive through all of the keys, that is a different deal, but to spend a whole week on key West is a lot of relaxing (or boredom depending on the person) after seeing the island. Not much to do other than sun, fish, snorkle, dive, eat, and drink. I personally love those things, but one that likes a lot to do might get bored in Key West after a day or 2.
Destin has beautiful beaches with soft powdery sugar white sand. The water is the turquoise that is mainly seen in the Caribbbean. The beach runs for miles and you can find a lot of secluded places where you have it almost to yourself. Lots of shops, outlet mall, restaurants, sea doo rentals, pontoon boat rentals, parasailing, water parks, tons of golf courses, some almost all night party spots (Baytowne Wharf), and a short drive to numerous other beaches (Panama City, Fort Walton, Seaside (where the Truman Story was filmed), Watercolour, Grayton Beach, Perdido key, Pensacola Beach, Gulf Shores, etc). Drive to Pensacola to see the Air and Space museum, Panama city to Shell island, Gulf Shores to the Flora-Bama. Mobile to the USS Alabama WW II battleship and also tour a submarine. Marine World has lots of animals, fish and dolphin shows. There is greyhound racing, and lots more within Destin or a short drive from there.
I love both places, but go to Destin often because it is only a 6 1/2 hour drive from my house. Key West is a location I have only been to a few times because it is so far. In the winter Key West is much better than Destin because Destin can get chilly. In the spring, summer, or fall the weather at either should be similar.