Did I miss something, what does it kill?

Fatty acids in most soaps - including shampoo and dish soap - work on soft bodied insects. It breaks down their skin . Soaps work only on contact and have no residual action .

Here is a portion of a UF article on use of natural products in the garden listing which insects are affected :

" Insecticidal soaps are made from plant oils (cottonseed, olive, palm, or coconut) or animal fat (lard, fish oil), but are generally not considered botanicals. They are made from the salts of fatty acids, which are in the fats and oils of animals and plants.

The mode of action is still unclear, despite years of use. Soaps are thought to physically disrupt the insect cuticle (outer skin), but additional toxic action is suspected. Soaps act on contact and must be applied directly to the insect to be effective. No residues remain on plants. They are effective against soft-bodied insects like aphids, some scales, psyllids, whiteflies, mealybugs, thrips, and spider mites. Hard-bodied insects (e.g., adult beetles or wasps) are not harmed because of their tough, chitinous bodies.

Some plants may be sensitive to soaps, resulting in leaf burn. Plants that have hairy leaves may be more susceptible to soap injury than smooth-leaved plants. Consult the label to see which plants are listed. Apply the soap spray on a small area of the plant to check for phytotoxicity. Commercial soaps are less likely to be phytotoxic. "