Generally “nuclear bomb/weapon” is the broad term that encompasses all weapons that derive their energy from nuclear bonds rather than chemical bonds
Atomic bombs generally refer to fission only devices like Fat Man and Little Boy that are solely powered by the fission of uranium or plutonium. These are big devices and are relatively weak (< 100 kT)
Thermonuclear/Hydrogen bombs get a boost from fusion of hydrogen(generally tritium which is 1 proton + 2 neutrons) and are significantly more destructive. Big Hydrogen Bombs like the Tsar Bomba get up to 50 MT but are massive devices(27 tons), the other end is stuff like the US W76 warhead which is only 100 kT but also only 95 kg so you can put it on a relatively small sub launched missile rather than having to use a massive bomber to drop it.
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