There are a few more meanings to stake, but assuming you are after the one that is closest related to stick I'd say a stick is a thin piece of wood that you'd use to play fetch with your dog and that often times just fell off the tree. A stake on the other hand would be bigger and can be made of metal or wood and is often hammered into the ground (eg cattle fence posts) or even some ancient huts on water are built on stake pillars.