Direct (solute-water) and indirect (water-water) contributions to adsorption at an air-water interface are identified using the Widom potential distribution theorem and quantified using molecular dynamics simulations of a liquid water droplet containing either neopentane or iodide-like solutes with charges of 0 or +/- 1. The results are used to quantitatively compare direct and indirect energetic and entropic contributions to adsorption, as well as to critically test surface capillary wave, linear response (LR), and mean field (MF) predictions. The negative signs of the total adsorption energies and entropies of both the anionic and cationic solutes are found to result from indirect adsorption induced changes in water-water interactions, rather than from surface capillary wave perturbations, which are found to be asymmetric with respect to solute charge. The LR and MF approximations both accurately describe the adsorption of neutral (hydrophobic) solutes, while for ionic solutes the MF approximation is entirely inappropriate and LR predictions are qualitatively (but not quantitatively) accurate.