Carbon nanotube (CNT) electrodes in supercapacitors have recently demonstrated enhanced performance compared to conventional carbon-based electrodes; however, the underlying relationships between electrode curvature and capacitance remain unclear. Using computer simulations, we evaluate the capacitive performance of metallic (6,6), (10,10), and (16,16) CNTs in [BMIM][PF6] ionic liquid (IL), with particular attention to the relative contributions of the electric double layer (EDL) capacitance (CD) at the CNT/IL interface and the electrode quantum capacitance (CQ). Our classical molecular dynamics simulations reveal that CD improves with increasing electrode curvature, which we discuss in terms of how the curvature affects both the electric field strength and EDL microstructure. In addition, the CQ of the CNTs is constant near the Fermi level and increases with curvature, as also demonstrated by density functional theory calculations. Our study shows that the electrode curvature effect on the total interfacial capacitance can be a strong function of applied voltage, which we attribute to the shifting contributions of CQ and CD.