The difference between the two kinetic energies is the total work done on the body. In your case the force pushing up stone on the hill will do some work. The gravity force will do the same work but with opposite sign - this time the displacement is in a direction opposite of the force. When you add the two amounts of work done by the two forces on the stone, they cancel each other and the total work is zero.
The key here is that you cannot talk about work done on a body, you always need to specify which force has done the work - you talk about work done no a body by a force. In the very special case where you consider the total work, which is the work done by all forces acting on the body, you can say that it is the difference between the initial and the final kinetic energy.