We have performed quantum-mechanical as well as classical magnetotransport calculations for a lattice of antidots with additional barriers. Both approaches reproduce the experimentally observed well-known commensurability peaks simultaneously with a negative differential magnetoresistance. By comparing both calculations, we find that quantum interference leads to an enhancement of the latter effect because of the increased probability of closed paths due to the insertion of additional potential barriers. We analyze the influence of the strength of inelastic scattering as well as increasing temperatures, i.e. energy averaging, in order to separate the classical and the quantum contributions.