A "sprout" is a seedling, the beginnings of a root, stalk or branch. Example: Did you see that my beans have sprouted? A "bud" will grow into a leaf or a flower. Example: Look at that rosebud! I'm sure that you will a beautiful red rose in a few days.

A sprout can mean a shoot or bud. A bud is an undeveloped shoot, leaf or flower, in the shape of a small knob.