Differential forms and exterior derivatives are used in Cartan's moving frame method, which allows one to calculate the curvature and Levi-Civita connection of a Riemannian manifold quite elegantly.

Normally the differential form calculations are shorter and less error prone than if one where to try to use Christoffel symbols.

The method is especially well adapted to cases where it's easy to find a section of the orthonormal frame bundle. Examples include: Conformal metrics (say the Poincaré metric), Thurston's NIL and SOL geometries on $\mathbb{R}^3$, and, more generally, invariant metrics on Lie groups.