We investigate the copper-wear-protective effects of graphene and boron nitride in single asperity sliding contact with a stiff diamond-coated atomic force microscopy (AFM)-tip. We find that both graphene and boron nitride retard the onset of wear of copper. The retardment of wear is larger with boron nitride than with graphene, which we explain based on their respective out-of-plane stiffnesses. The wear protective effect of boron nitride comes, however, at a price. The out-of-plane stiffness of two-dimensional materials also determines their friction coefficient in a wear-less friction regime. In this regime, a higher out-of-plane stiffness results in larger friction forces.