wood:
inexpensive, easy to balance (sand), "gives" first when striking an object (ground, pole, fingers )

synthetic:
generaly more effecient, more consistant from prop to prop than wood. doesnt break as easy as wood (can damage crankshaft, fingers, often called whirling ginsu knives)

this is the basic diffrences. mostly you want to go with composite if you are trying to get that last bit of performance out of the plane, otherwise wood is good enough for sport flying.