The only difference for our purposes is predictability. When you buy a Labrador, you know, generally, how big he'll be and how much coat he'll have. Mixed breeds are a kaleidoscope of their special combination, history, and environment. If you don't mind a bit of mystery in your life, get a mixed-breed pup. He'll greet you at the door, bark at the stranger on your porch, and fetch a ball for your kids just as well as any purebred.
Puppies are cute, demanding, and unfettered by any understanding of human rules.
Puppies are usually not the best choice for a household with children under the age of five. Adult dogs are often housebroken and past the intense chewing stage. On the down side, bad habits are sometimes firmly in place. Old dogs can learn new tricks, though, so don't rule them out. The older dog is a fine choice for many families.