This paper presents a systematic comparative analysis of the Finite Difference Method (FDM) and the Finite Element Method (FEM) as applied to the numerical solution of partial differential equations (PDEs) arising in engineering and applied science. Both methods are examined with respect to their theoretical foundations, discretization strategies, convergence properties, and computational performance. Benchmark problems including the Poisson equation, transient heat conduction, and linear elasticity are solved using both approaches under controlled conditions. Numerical experiments demonstrate that FDM offers lower implementation complexity and computational cost on structured domains, while FEM provides superior flexibility, accuracy, and adaptivity for complex geometries. Quantitative performance indicators — including error norms, convergence rates, CPU time, and memory usage — are tabulated and discussed. The findings offer practical guidance for selecting the appropriate method based on problem characteristics.