In general, a bud is a specialised structure in the organisms which help in vegetative propagation.
1.The bud in the case of Bryophyllum grows on the leaf and fall to give rise to a independent plant while in case of Hydra the bud formation start at the budding zone. It increases in diameter and elongates and then detaches to live a independent life.
2.The Bryophyllum has much larger number of buds as compared to the Hydra.
3.The buds of the Bryophyllum remain in dormant state until they come in contact with soil while the bud in hydra is always active.