Using first-principles calculations within density functional theory, we study the relative impacts of quantum confinement and strain on the electronic structure of two II-VI semiconductor compounds with a large lattice-mismatch, CdSe and CdTe, in core/shell nanowire geometries with different core radii and shell thicknesses. For fixed CdSe core radius, we find that the electronic band gap in the core is significantly reduced with increasing CdTe shell thickness, by an amount comparable to that expected from quantum confinement, due to the development of a large and highly anisotropic strain throughout the heterostructure. A straightforward analysis allows us to separate quantitatively changes in band gap due to quantum confinement and strain. Our studies elucidate and quantify the importance of strain in determining the electronic and optical properties of core/shell nanostructures.