Have you tried to actually write an integrator in excel? It gets rather cumbersome. Being able to name things, call functions, and run loops is so much nicer in an actual programming language compared to excel.
using the backward difference method to calculate dx/dt and d2x/dt2 based off the previous time step
In short, you're just using a different integration scheme. You've implemented a low-order implicit scheme it sounds like.
Try writing a couple of other integrators like euler, modified euler, and maybe RK4 if you've got the patience and/or vba skill to make sure it's typed in properly. Plot the results, errors, etc to see the differences. Maybe compute the spectral response if you're feeling ambitious!