Warning: The first paragraph of the following is outside my expertise.
I am told this construction is very useful in PDE's. If you have a PDE on some manifold $M$, you can often formulate the vector space of solutions as the kernel of some flat connection on a vector bundle. In particular, I believe that the analytic side of the Atiyah-Singer index theorem is the Euler characteristic of the deRham theory you have described.
I can tell you that the analogous construction is very important in complex algebraic geometry. Given a holomorphic vector bundle on a complex manifold, there is a natural way to define an anti-holomorphic a $d$-bar connection on it. (This mean $\nabla_X$ is only defined when $X$ is a $(0,1)$ vector field.) The cohomology of the resulting "deRham deRham-like complex" , which is then called the Doulbeaut complex in this setting, is the same as the cohomology of the sheaf of holomorphic sections of the vector bundle. See Wells' Differential Analysis on Complex Manifolds or the early parts of Voisin's Hodge Theory and Algebraic Geometry.
