I believe that "nuclear bomb" is a generic term for any bomb that utilizes nuclear energy. There are two basic types of nuclear energy: fission and fusion.

In nuclear fission a radioactive heavy atom (more protons than an iron atom), like certain isotopes of uranium or plutonium, decays, or breaks apart into lighter atoms, releasing some energy in the process. Nuclear fusion involves combining two or more light atoms (fewer protons than an iron atom), such as hydrogen or helium, to create a heavier atom which also releases energy.

According to my dictionary, an atomic bomb is a nuclear weapon that uses a nuclear fission explosion primarily. The more powerful hydrogen bomb, also known as an H-bomb, fission bomb, or a thermonuclear bomb, uses nuclear fusion of tritium, a hydrogen atom that has two neutrons in addition to the one proton. Because a fusion explosion is difficult to get started, a hydrogen bomb actually uses nuclear fission to provide enough heat to get the process going.

There is really no difference. People use both names. Nuclear bomb is the more recent and more correct term since it is in the nucleus of the atom that changes take place which are associated with vast amounts of energy being released.