In this work, we compare the friction behavior of single asperity contacts of a PtSi tip with copper and monolayered graphene on copper. For both PtSi/copper-and PtSi/graphenized copper tribological couples, we measured 16 differently oriented grains with three different sliding velocities as a function of the load by atomic force microscopy to assess the role of crystallographic orientation on friction and its rate dependence. We find that friction on copper is governed by shearing and does not depend on the crystallographic orientation but is significantly affected by the sliding velocity. We discuss this dependence based on contact aging in ambient conditions and shear strain rate sensitivity. In contrast, we find that friction of a graphene monolayer on copper is governed by puckering and depends both on the crystallographic orientation of the underlying grain and on the sliding velocity. We discuss these dependencies based on the orientational commensurability of the graphene lattice with the underlying copper crystallographic plane and on thermolubricity. Beyond their validity at the nanometer scale, our results are relevant to extend the understanding of friction between engineered surfaces at the macro- and micro-scale, where contacts initiate at nanometer-scale asperities.