Thermal boundary conductance is typically positively correlated with interfacial adhesion at the interface. Here, we demonstrate a counterintuitive experimental result in which a weak van der Waals interface can give a higher thermal boundary conductance than a strong covalently bonded interface. This occurs in a system with highly mismatched vibrational frequencies (copper/diamond) modified by a self-assembled monolayer. Using finely controlled fabrication and detailed characterization, complemented by molecular simulation, the effects of bridging the vibrational spectrum mismatch and bonding at the interface are systematically varied and understood from a molecular dynamics viewpoint. The results reveal that the bridging and binding effects have a trade-off relationship and, consequently, that the bridging can overwhelm the binding effect at a highly mismatched interface. This study provides a comprehensive understanding of phonon transport at interfaces, unifying physical and chemical understandings, and allowing interfacial tailoring of the thermal transport in various material systems.