This research was based on a nano-AAT (American Advanced Technology)-modified asphalt to which CRP (crumb rubber powder), a rubber waste, was introduced to explore the influence of CRP on AAT performance. The changes in the performance of AAT-modified asphalt after the addition of CRP were analyzed. The rubber powder with the raw material of waste tire was added to the asphalt instead of SBS modifier. While achieving waste recycling, the asphalt material has good performance. Physical analysis methods, rheological performance tests, rolling thin-film oven tests and Fourier transform infrared spectroscopy tests were used to investigate the performance of the composite-modified asphalt. The rheological properties of the composite-modified asphalt were analyzed by means of DSR, BBR and MSCR tests, and the microscopic mechanism of the modified asphalt was investigated by means of FTIR tests. The optimal nano-AAT-composite-modified formulation A3C3 (AAT-3.5%SBS-3%CRP) was selected by evaluating the overall performance. Additionally, the performances of the AAT/CRP-composite-modified asphalt and SBS-modified asphalt were compared using physical indicators, the rutting factor, creep flexibility and the stiffness modulus. The results show that the A3C3-modified asphalt had better stiffness, high-temperature (HT) performance and aging resistance than the SBS-modified asphalt, but it was less effective at low temperatures (LTs). According to FTIR, the absorption curves of A3C3 and SBS are essentially equal, with A3C3 only having a variation at 1104 cm−1.