Disclaimer: This is almost definitely a deeply flawed explanation. But, it helps me to think about it this way.
Nuclear weapons get their explosive force by splitting heavy atoms (Fission) or merging light atoms (Fusion). Both processes rely on the conversion of physical matter into energy. The trick is: when two atoms are fused, the resulting atom is lighther than the sum of it's parts. This mass can't simply disappear: the extra mass is converted into an explosion.
Fission uses heavy elements that have absorbed lots of energy (and lost mass) when they were formed millions of years ago in exploding stars. During fission (inside the bomb / millions of years later) this energy is simply released when the atom is split.
Fusion uses light elements that actually release energy when they fuse together. They don't take much energy to bind, so the lost mass doesn't all get used up in the binding energy - the extra mass gets blasted out into the world as the nuclear explosion.
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