Answer
Both buckets weigh the same.
Work Step by Step
Since the piece of wood is in equilibrium, the buoyant force on the wood is equal to the wood's weight. According to Archimedes' Principle, the buoyant force is equal to the weight of the water that is displaced by the wood.
Therefore, it does not make any difference if there is a piece of wood or not. The weight of the wood is equal to the weight of the water that is displaced, so the total weight in each bucket is the same.
