For someone who does not already know Lie group and bundle theory, the formulation of covariant derivatives through parallel transport in the principal, and associated vector bundles, might seem unnecessarily complicated. In that light, I wondered what the virtues of the principal/associated bundle formulation is? In particular what does one get from this formulation, that one does not get from the formulation of the covariant derivative as a map from smooth vector fields to smooth vector fields on a manifold M? Does it give some new insights? Or is it somehow more general?