We demonstrate, using molecular-dynamics computer simulations, the strong influence of surface wettability on the equilibrium structure of the electrical double layer at solid interfaces and on electrokinetic transport in aqueous electrolytes due to the effects of interfacial ion specificity and hydrodynamic slip. In particular, we show that anomalous electrokinetic effects such as nonzero zeta potentials for uncharged surfaces are general features of electro-osmotic flow in hydrophobic channels for electrolytes with substantial cation/anion size asymmetry, as a result of the stronger attraction of the larger ion to the "vapor-liquid-like" interface induced by a hydrophobic surface. We establish that the simulated velocity profiles obey continuum hydrodynamics on the nanoscopic length scales studied and show that the anomalous flow profiles can be accurately predicted by using a modified Poisson-Boltzmann description for the ion density distributions that incorporates an ion-size-dependent hydrophobic solvation energy as a crucial component. We also demonstrate that, counterintuitively, the flow for a charge-neutral fluid is independent of the solid-fluid friction coefficient.