In the finite element method (FEM) the domain is divided into a finite number of subdomains. Therefore, also the name of the method. For each subdomain local functions are defined that should approximate the solution of the problem under consideration. The derivation of the FEm is now based on the weak form of the equilibrium equations. From the partial differential equations we go to an integral formulation. Since the continuity requirements for the functions describing the solution are lower in the integral formulation it is called weak form.
In the finite difference method (FDM) the discretization starts directly with the partial differential equations. In this case alle derivatives are replaced by suitable difference formulae. Here you can distinguish between backward, forward and centered differences.