The difference between an atomic bomb and a nuclear bomb is that "nuclear bomb", is mainly a broader term that is used to describe all types of weapons which utilize nuclear reactions for the explosive power of the weapons which also include atomic bonds.
Atomic bombs are also known as fission bombs and use nuclear fission, which is the splitting of heavy atomic nuclei like uranium or plutonium releases energy.
Nuclear bombs can also include other types of nuclear weapons which use fusion or even a combination of fission and fusion and can also include hydrogen bombs or thermonuclear bombs.
Atomic bombs are bombs that rely soley on the process of nuclear fission, which is when heavy atomic nuclei such as uranium or plutonium split apart and release tremendous energy.
Nuclear bomb is a more broader term that encompasses all types of nuclear weapons which include fission bombs or also atomic bombs, fusion bombs or hydrogen bombs and those bombs that use a combination of both.
Hydrogen bombs are bombs which use nuclear fusion, in which light atomic nuclei such as hydrogen fuse together to form heavier nuclei and release even more energy than fission.
Hydrogen bombs are often much more powerful than fission bombs are.
Nuclear bombs, and most specifically thermonuclear bombs (hydrogen bombs), are more powerful than atomic bombs (fission bombs).
Thermonuclear bombs achieve their greater destructive power by combining fission and fusion reactions, releasing significantly more energy than atomic bombs, which only use fission.