So bottom line, atomic and hydrogen bombs are terms used and derived from history more than from physics. However, the difference between fission and fusion bombs, which are better terms for the two kinds, is straightforward. Fission bombs split heavy elements into lighter ones to get the energy. Fusion bombs merge lighter elements into heavier ones to get the energy.
In both cases, fusion or fission, the energy is derived from the mass deficiency that occurs during the fission or fusion. In both cases, the mass before the reaction is greater than the mass after the reactions. And that difference, the deficiency, is where the energy comes from as in e = dM c^2, where dM is the amount of mass deficiency.
Fusion bombs are way more powerful than fission bombs. In fact, to get the hydrogen isotopes to fuse, a fusion bomb uses a small fission bomb to create the heat and pressure needed for the fusion process. But even the most powerful fusion bomb ever exploded, the 50 MT Tsar Bomba of the former Soviet Union, was just barely a blip on the USGS seismographic scale. Compared to the recent 9.0 earthquake off Japan, your H-bomb is a mere nat on Mother Earth's skin.