Although both atomic and nuclear bombs derive their immense power from reactions that take place in the nucleus of the atom, they do so in different ways. As the terms are most commonly used, an atomic bomb derives its energy from fission, or splitting atoms, while a nuclear device, or "H-bomb," is driven by fusion.
The physics behind the two weapons is similar, according to About.com. In an atomic bomb, one heavy atom, usually enriched uranium or plutonium, is split and emits enough neutrons to split more atoms. This releases tremendous energy that is usually measured in the equivalent destructive power of thousands of tons of TNT, or kilotons.
Nuclear bombs use atomic fission as a trigger for a still more energetic explosion. The high temperatures generated by plutonium fission encourage atoms of hydrogen to fuse into helium. Again, this releases energy in an uncontrolled way over the target. Although the fusion reaction is, measure for measure, less efficient than fission, hydrogen is very light and can be packed densely into the core of a nuclear bomb. The yield of hydrogen bombs is orders of magnitude higher than fission alone can generate, and it is measured in megatons, or millions of tons of TNT.