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.