Differences between "atomic bomb" and "nuclear weapon"
Solution 1:
Atom bombs rely solely on fission; thermonuclear weapons rely on both fission and fusion. With so many new ways to deliver them (including a cannon mounted on a jeep), we left the word bomb behind in favor of weapon.
Nuclear Weapons, Wiki excerpt:
Fission weapons (A-Bombs)
All existing nuclear weapons derive some of their explosive energy from nuclear fission reactions. Weapons whose explosive output is exclusively from fission reactions are commonly referred to as atomic bombs or atom bombs (abbreviated as A-bombs). This has long been noted as something of a misnomer, as their energy comes from the nucleus of the atom, just as it does with fusion weapons.
Fusion weapons (H-Bombs)
The other basic type of nuclear weapon produces a large proportion of its energy in nuclear fusion reactions. Such fusion weapons are generally referred to as thermonuclear weapons or more colloquially as hydrogen bombs (abbreviated as H-bombs), as they rely on fusion reactions between isotopes of hydrogen (deuterium and tritium). All such weapons derive a significant portion, and sometimes a majority, of their energy from fission. This is because a fission weapon is required as a "trigger" for the fusion reactions, and the fusion reactions can themselves trigger additional fission reactions.
Solution 2:
Within the military, Atomic specifically refers to fission-only weapons. Thermonuclear refers to any weapon with fusion somewhere in the explosive process. Nuclear is the generic term for both, where the main energy release mechanism comes from changes in the nuclear forces of atoms (rather than chemical forces, like conventional - non-nuclear - explosives). The military also use the term Special Weapons to specifically refer to Nuclear Weapons. Gadget is also heard informally, as this was the informal code name for the original nuclear weapon during the Manhattan Project.
A bomb is unpowered. Among the trained in a formal setting, it will refer to something dropped from an aircraft, although it can be used in a non-formal setting, even by the trained, for anything that goes bang. Formally, a terrorist 'bomb' is referred to as an Improvised Explosive Device, or IED. The whole device, whether rocket, bomb, missile or whatever, is referred to as the weapon. The 'bit that goes bang' in a more complex weapon, such as a cruise missile, is referred to as the warhead. It does not have to be, and often isn't, at the head or front of the weapon. The nuclear element within a nuclear warhead is the physics package. It is inside the conventional explosives used to trigger it.
Thus atomic bomb is quite specific, and nuclear weapon very general.