Classical capillary theory predicts that a non-neutrally wetting ellipsoidal particle adsorbed at a liquid-vapor interface will deform the interface. The deformation gives rise to anisotropic capillary forces of a quadrupolar nature that induce strong directionality in the particle interactions. Here, we investigate the interactions between nanoparticles with characteristic lengths of 1-5 nm. We show that the near-field interactions are dominated by solvent-mediated forces, which arise from the fluid packing between the nanoparticles and direct nanoparticle-nanoparticle interactions. The solvent-mediated forces are two orders of magnitude larger than the estimated capillary force. We find that interacting ellipsoidal nanoparticles adsorbed at the liquid-vapor interface have a larger repulsion in the depletion region than the nanoparticles submerged in a dense bulk phase and argue that this is because of a negative line tension associated with the three-phase line.