This paper presents a comprehensive comparative analysis of two optimization algorithms: the widely-adopted Adam optimizer and the recently proposed MuonClip algorithm. Optimization algorithms play a crucial role in deep learning, directly influencing both convergence speed and final model performance. Through systematic comparison of mathematical structures, computational complexity, convergence characteristics, and empirical performance, we analyze the strengths and limitations of each algorithm. Our findings suggest that MuonClip offers superior memory efficiency and stability, while Adam provides finer adaptivity and broader applicability. These insights provide practical guidelines for algorithm selection in different scenarios.