The one thing the students never seem to bring up is that it was a Democrat that used this weapon of mass destruction. I doubt that most college students could tell you what party Harry Truman belonged to.
I doubt that most college students could tell you who Harry Truman WAS.
These are not exclusive categories. "Atomic bombs" (fission devices) are a category of nuclear weapons, the other existing category of nuclear weapons being fusion or thermonuclear devices ("hydrogen bombs").
Fission bombs are more correctly referred to as "fission-only" since fusion devices use an embedded fission device to initiate the fusion reaction.
Other types of nuclear weapons are theoretically possible, notably anti-matter devices.
All are "nuclear" in the sense that their energy is derived from the conversion of atomic mass (which is mostly in the nucleus) into energy.