The people of the Manhattan Project used the term atomic bomb internally and externally for their invention. "Atomic" was widely used as a synonym for "nuclear" in both lay communication and technical literature for at least a generation afterward. The International Atomic Energy Agency is still called such and never felt the need to correct this imagined terminological mistake.
That's not what he's referring to, though. In chemistry, bonds between "atoms" are ordinary chemical bonds, held together through electromagnetic forces, and yielding typical chemical reactions like explosions and deflagrations. Fission and fusion are "nuclear" processes, operating between protons and neutrons within the nucleus of an atom only, and moderated by the strong nuclear force. The energy released by breaking these bonds is orders of magnitude greater, and what we normally refer to as a "nuclear" explosion. Calling it "nuclear" vs. "atomic" conveys useful information about what's actually going on when it detonates.
