Are the terms thermonuclear and nuclear the same, or are they different (and if so how)

Actually thermonuclear some from the fact that in order to have fusion you can heat the fusion material to high temperature, so high that the nuclei overcome their repulsion forces and fuse together. Since the energy was supplied in the form of heat (though not by literally burning fire) it is thermal. Also it means that there are ions and electrons from the hydrogen and deuterium with a spread in their velocities, meaning with all possible kinetic energies.

The other fusion would be called a beam fusion where, rather than having a spread of energy of fusing protons you just accelerate a bunch of them to a single but very high energy. Beam fusion is not considered a viable source of energy, the thermonuclear on the other hand is.

Thermonuclear is using the HEAT which is created by a nuclear process.

The nuclear processes themselves may be used for other purposes and in other ways.