With Finite Differences, we discretize space (i.e. we put a grid on it) and we seek the values of the solution function at the mesh points. We still solve a discretized differential problem.
With Finite Elements, we approximate the solution as a (finite) sum of functions defined on the discretized space. These functions make up a basis of the space, and the most commonly used are the hat functions. We end up with a linear system whose unknowns are the weights associated with each of the basis functions: i.e., how much does each basis function count for out particular solution to our particular problem?