What is the difference between the Finite Difference and Finite Element Approach?

The short answer is that we use step functions for the finite difference solution and continuous functions for the finite element solution (the simplest being linear functions). So when you think of the parameters of a FD model you can think of them as
_|   |_

whereas you would think of the FE model as


In the end you have the same gross structure of matrix equations and invert the matrix to solve for the dependent variables
[A] {h} = {R}

Here is a paper by Mattiussi, 1997, from the Journal of Computational Physics with an in depth discussion