The Zen answer is that there is no difference.
Perhaps a more useful answer is that they are different techniques to achieving the same goal.
Mindfulness, also called shikantaza, (sometimes likened to the Yogic practice of Pratyahara) is the practice of focusing awareness on acknowledging stream of consciousness without judging or becoming absorbed in any one thought.
A koan is a sort of nonsense-riddle that is supposed, when pondered long enough, to cause a sudden insight. “What did your face look like before you were born?”