The differential capacitance of an electrical double layer is a topic of great importance to develop more efficient and environment-friendly energy storage device. In addition to electrostatic interactions, several theoretical and experimental works have suggested that the differential capacitance is affected by a complex interplay of ion-specific effects and solvent-mediated hydration interactions. In line with this, in this work we present a theoretical model that incorporates these contributions into the Poisson-Boltzmann framework by means of ion-specific  soft Yukawa potentials and non-ideal entropic contributions. More precisely, we focus on a flat planar surface (electrode) exposed to an electrolytic solution of uniform dielectric constant; in such a scenario, we use the Mean-Field approximation and Monte Carlo Simulations to analyze how the differential capacitance is influenced by electrostatic and hydration interactions, and by ions of unequal sizes. We also investigate the role played by the salt concentration on the so-called camel-shape to bell-shape transition.