General properties of the antagonistic biomolecular interactions between viruses and their hosts (exogenous interactions) remain poorly understood, and may differ significantly from known principles governing the cooperative interactions within the host (endogenous interactions). Systems biology approaches have been applied to study the combined interaction networks of virus and human proteins, but such efforts have so far revealed only low-resolution patterns of host-virus interaction. Here, we layer curated and predicted 3D structural models of human-virus and human-human protein complexes on top of traditional interaction networks to reconstruct the human-virus structural interaction network. This approach reveals atomic resolution, mechanistic patterns of host-virus interaction, and facilitates systematic comparison with the host's endogenous interactions. We find that exogenous interfaces tend to overlap with and mimic endogenous interfaces, thereby competing with endogenous binding partners. The endogenous interfaces mimicked by viral proteins tend to participate in multiple endogenous interactions which are transient and regulatory in nature. While interface overlap in the endogenous network results largely from gene duplication followed by divergent evolution, viral proteins frequently achieve interface mimicry without any sequence or structural similarity to an endogenous binding partner. Finally, while endogenous interfaces tend to evolve more slowly than the rest of the protein surface, exogenous interfaces--including many sites of endogenous-exogenous overlap--tend to evolve faster, consistent with an evolutionary "arms race" between host and pathogen. These significant biophysical, functional, and evolutionary differences between host-pathogen and within-host protein-protein interactions highlight the distinct consequences of antagonism versus cooperation in biological networks.