In the existing models for finite non-cooperative games on networks, it is usually assumed that in each single round of play, regardless of the evolutionary update rule driving the dynamics, each player selects the same strategy against all of its opponents. When a selfish player can distinguish the identities of its opponents, this assumption becomes highly restrictive. In this paper, we introduce the mechanism of strategic differentiation through which a subset of players in the network, called differentiators, are able to employ different pure strategies against different opponents in their local game interactions. Within this new framework, we study the existence of pure Nash equilibria and finite-time convergence of differentiated myopic best response dynamics by extending the theory of potential games to non-cooperative games with strategic differentiation. Finally, we illustrate the effect of strategic differentiation on equilibrium strategy profiles by simulating a non-linear spatial public goods game and the simulation results show that depending on the position of differentiators in the network, the level of cooperation of the whole population at an equilibrium can be promoted or hindered. Our findings indicate that strategic differentiation may provide new ideas for solving the challenging free-rider problem on complex networks.