In this paper, we consider Simultaneous Transmission and Reflection (STAR) Reconfigurable Intelligent Surface (S-RIS) and passive RIS (P-RIS) assisted integrated sensing and communication system (ISAC), where S-RIS is enabled to broadcast communication signal, and P-RIS assists sensing functionalities. In particular, we jointly optimize the beamforming vector at the multi-antenna ISAC transmitter, and phase shift vector to maximize the weighted sum-rate (WSR) at the communication users while taking care of the maximum power limit at ISAC transmitter while ensuring the performance of sensing model to detect targets in its vicinity and limitations of phase and amplitude of S-RIS elements. To address the non-convexity of the above problem, we propose a low-complexity alternating optimization (AO) algorithm. Furthermore, we provide a comprehensive simulation-based graphical results to verify the viability of the proposed framework with its P-RIS assisted counterpart. Eventually, exhaustive simulation results are demonstrated to present the impact of RIS elements and the number of antennas at the ISAC transmitter. Accordingly, we illustrate the impact of S-RIS and the number of targets to highlight the trade-off between sensing and communication.