In piaffe, it is only the carrying that is activated in the hindquarter. The horse moves almost in place.
In passage, both the carrying and pushing are present in the hindquarter. The horse moves itself forward-upwards.
In piaffe, the horse brings its hind legs under his point of weight with bended, flexible and supple joints. The addition of some push leads to a transition into passage or extended trot.
In passage, the push is aimed in a forward-upward direction (green arrow). In extended trot, the push is directed more backwards (red arrow).