There is indeed a subtle difference between the two. Both derive their destructive power from a nuclear reaction, but there are two main types of bomb - those that use a nuclear fission reaction, and those that use a more powerful nuclear fusion reaction.
The nuclear fission devices are known as atomic bombs, atom bombs, a-bombs, and are of the type used in world war 2, and composed much of the early arsenal of "special weapons" post ww2.
The nuclear fusion devices are much more powerful (sometimes up to 1000 times more powerful) than fission devices, and are known as hydrogen bombs, h-bombs, fusion bombs, or thermonuclear devices ("nukes" for short).
In conclusion then, all devices are based on "nuclear" reactions, but the term "atomic bomb" usually refers to the more antiquated nuclear fission based devices.
Hope that helps a little.