The large thermal resistance across the carbon nanotube (CNT)-polymer matrix interface is a limiting factor for achieving polymer composites with high thermal conductivities. Using equilibrium molecular dynamics simulations we show that an azide-terminated aromatic polymer HLK5 (C22H25O3N3) functionalized onto the CNT sidewall can efficiently decrease the thermal resistance between the nanotube and different types of polymer matrices (polystyrene, epoxy, and polyethylene). The HLK5 functionalization can also significantly decrease the CNT-CNT junction resistance. Compared with hydroxyl and octane functionalizations, the HLK5 one alters less the high intrinsic CNT thermal conductivity at the same surface coverage ratio. By revealing the important role played by the atomistic van der Waals interactions in attaining these key results, our study brings a new perspective in the nanoscale design of advanced CNT-polymer materials (Graph Presented).