  We present a study of the structure and differential capacitance of electric
double layers of aqueous electrolytes. We consider Electric Double Layer
Capacitors (EDLC) composed of spherical cations and anions in a dielectric
continuum confined between a planar cathode and anode. The model system
includes steric as well as Coulombic ion-ion and ion-electrode interactions. We
compare results of computationally expensive, but "exact", Brownian Dynamics
(BD) simulations with approximate, but cheap, calculations based on classical
Density Functional Theory (DFT). Excellent overall agreement is found for a
large set of system parameters $-$ including variations in concentrations,
ionic size- and valency-asymmetries, applied voltages, and electrode separation
$-$ provided the differences between the canonical ensemble of the BD
simulations and the grand-canonical ensemble of DFT are properly taken into
account. In particular a careful distinction is made between the differential
capacitance $C_N$ at fixed number of ions and $C_\mu$ at fixed ionic chemical
potential. Furthermore, we derive and exploit their thermodynamic relations. In
the future these relations are also useful for comparing and contrasting.
