Originally posted by Marc Benford
The title pretty much says it all.

A puzzle is generally something that could occur in a real game. They are tactics exercises designed to prepare you for what you will see in real games.

A problem is generally an artistic construction that uses the game of chess as its medium. In problems, there are different conventions. Ideally, the position should not have ANY unnecessary pieces (economy). Composers generally don't care if the position is not that likely to occur in a real game, as long as the position is still legal. Also, the goal of the game often changes in a chess problem. For examples, there are helpmates, in which white and black collaborate to achieve checkmate of black.