In this study, a comparative analysis of three widely used numerical methods for solving differential equations is presented: the Finite Difference Method, the Finite Element Method, and Spectral Methods. The mathematical foundations, algorithmic structures, and implementation aspects of each method are thoroughly investigated using MATLAB. Key parameters influencing the accuracy, convergence behavior, and computational efficiency of each approach are identified. The performance of the methods under various boundary conditions and geometric configurations is examined. Criteria for selecting the most appropriate method based on problem complexity and required accuracy are defined. The limitations and advantages of each technique are established through a theoretical framework. A structured comparative table is proposed, summarizing critical attributes such as geometric flexibility, solution accuracy, convergence rate, and practical applicability. A unified conceptual basis is developed to support the effective selection of numerical methods for real-world modeling problems. Simulation results and graphical comparisons are presented to validate the theoretical findings and demonstrate the practical implications of each approach.