Current autonomous driving systems often struggle to balance decision-making and motion control while ensuring safety and traffic rule compliance, especially in complex urban environments. Existing methods may fall short due to separate handling of these functionalities, leading to inefficiencies and safety compromises. To address these challenges, we introduce UDMC, an interpretable and unified Level 4 autonomous driving framework. UDMC integrates decision-making and motion control into a single optimal control problem (OCP), considering the dynamic interactions with surrounding vehicles, pedestrians, road lanes, and traffic signals. By employing innovative potential functions to model traffic participants and regulations, and incorporating a specialized motion prediction module, our framework enhances on-road safety and rule adherence. The integrated design allows for real-time execution of flexible maneuvers suited to diverse driving scenarios. High-fidelity simulations conducted in CARLA exemplify the framework's computational efficiency, robustness, and safety, resulting in superior driving performance when compared against various baseline models. Our open-source project is available at https://github.com/henryhcliu/udmc_carla.git.