The difference of layering and budding is:
Layering- the branch is bent down to the ground and covered with soil with its tip exposed. Roots will then grow from the covered branch, and a shoot grows from the exposed tip. The shoot( new plant) can then be cut off from the mother plant.
Budding- it is also like grafting. A bud from the branch of the desired plant is slipped into a T-shaped slit in the bark of another tree. The stem just above the bud graft is then cut off, making the bud graft the main stem of the tree.
Hope it helps.