Bitumen exhibits viscoelastic properties, showcasing both viscous and elastic behaviors, which are characterized by the phase angle (Ø) and dynamic modulus (|E*|). Issues like early fatigue fractures, rutting, and permanent deformations in bituminous asphalt pavements arise due to moisture susceptibility, high-temperature deformation, low-temperature cracking, and overloading. These distresses result in potholes, alligator cracks, and specific deformations that lead to early pavement failure, increasing rehabilitation and maintenance costs. To address these issues, this study examines the dynamic response (|E*|) and phase angle (Ø) behavior of Styrene Butadiene Styrene (SBS) modified and unmodified asphalt mixtures. SBS was incorporated in various proportions, ranging from 2–7% by the weight of bitumen. The asphalt mixture performance tester (AMPT) was utilized to measure the |E*| at temperatures of 4.4, 21.1, 37.8, and 54.4°C, and frequencies of 0.1, 0.5, 1, 5, 10, and 25 Hz. The study found significant correlations between dynamic modulus, temperature, loading frequency, and SBS content. Additionally, Multi Expression Programming (MEPX) and regression modeling were employed to estimate the |E*| of SBS-modified HMA. Results indicated that increasing SBS content up to 7% decreased penetration and ductility values by up to 46% and 56%, respectively, while raising the softening point by 63% due to increased stiffness. The blend with 6% SBS by weight of bitumen exhibited superior performance compared to other mixtures. Phase angle initially increased with rising temperature, peaking at 37.8°C at lower frequencies, and continued to increase at higher frequencies. Isothermal and isochronal plots showed that the 0% SBS mix had a higher phase angle due to increased bitumen content. Overall, the HMA mix with 6% SBS provided the best outcomes.