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It’s apparently a common misconception that there’s a difference between the two, but there really isn’t. A “hydrogen bomb” is a regular ole’ implosion fission device like Fat Man was, but the initial nuclear fireball is used to heat and compress a “secondary” element, typically containing substances like deuterium and tritium (both of which are hydrogen isotopes) and lithium, which undergoes fusion. The high temperatures and high neutron flux created by this fusion reaction typically in turn initiates fission in a fissionable (but not fissile) outer shell, usually U-238…which would not undergo fission without the added heat and neutrons of the fusion reaction. So in most (all?) designs of “hydrogen” bombs, the majority of the power comes from fission, not fusion.
Anyway, with the advent of “boosting”, even weapons which do not have a fusing secondary (i.e. single stage warheads, of typically small size and comparatively small yield) utilize fusion by injecting a small amount of tritium gas (again, an isotope of hydrogen) into the center of the fission core right before detonation. The tritium undergoes fusion at the heart of the explosion and creates a huge amount of neutrons, which then cause more of the fission material to actually fission…material that would otherwise just be blown apart in the blast without actually reacting. The vast majority of Pu or U in a nuclear weapon is “wasted” in that it is never converted to energy and doesn’t contribute to the size of the explosion. Boosting increases the amount of Pu or U which does convert to energy, and this significantly increases the yield per mass–which means you can make either a bigger explosion with the same amount of plutonium or uranium, or make the same-sized blast with less.
These days every weapon is “at least” a boosted weapon, so in that sense every nuke is a “hydrogen bomb.”