An atom bomb is a fission device. Here a heavy atom (such as uranium) is split into two or more lighter atoms. In such a nuclear fission process, one or two neutrons are also released from each atom. If these neutrons are captured by the surrounding heavy atom, further fission takes place leading to further neutron release. In atomic power reactors, this chain reaction is controlled so that the energy released during the reaction can be used beneficially.
But in the case of an atom bomb, the chain reaction is not controlled but left free. This leads to the release of large amounts of heat energy in a very short time resulting in a catastrophic explosion.
In a fusion bomb, such as the hydrogen bomb, two or more lighter atoms fuse together to form a heavier atom. Such fusion reactions can take place only at very high temperatures; say a few million degrees Celsius.
Only at such temperatures can the starting atoms overcome the repulsive forces acting between them.
In this case also, when uncontrolled fusion takes place a large amount of energy is suddenly released resulting in an explosion far bigger than an atomic explosion.
It may be noted that the temperature needed to initiate a fusion reaction is normally obtained by a fission reaction.