Nuclear weapons use fission as either the partial or the main energy source.  Depending on the weapon design and where it is exploded, the relative importance of the fission product radioactivity will vary compared to the activation product radioactivity in the total fallout radioactivity.
The immediate fission products from nuclear weapon fission are essentially the same as those from any other fission source, depending slightly on the particular nuclide that is fissioning.  However, the very short time scale for the reaction makes a difference in the particular mix of isotopes produced from an atomic bomb.
For example, the Cs/Cs ratio provides an easy method of distinguishing between fallout from a bomb and the fission products from a power reactor. Almost no caesium-134 is formed by nuclear fission (because xenon-134 is stable). The Cs is formed by the neutron activation of the stable Cs which is formed by the decay of isotopes in the isobar (A = 133). So in a momentary criticality, by the time that the neutron flux becomes zero too little time will have passed for any Cs to be present. While in a power reactor plenty of time exists for the decay of the isotopes in the isobar to form Cs, the Cs thus formed can then be activated to form Cs only if the time between the start and the end of the criticality is long.
According to Jiri Hala's textbook, the radioactivity in the fission product mixture in an atom bomb is mostly caused by short-lived isotopes such as iodine-131 and barium-140. After about four months, cerium-141, zirconium-95/niobium-95, and strontium-89 represent the largest share of radioactive material. After two to three years, cerium-144/praseodymium-144, ruthenium-106/rhodium-106, and promethium-147 are responsible for the bulk of the radioactivity. After a few years, the radiation is dominated by strontium-90 and caesium-137, whereas in the period between 10,000 and a million years it is technetium-99 that dominates.