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Inspired by Ulrich Pennig's answer, I'll mention that Chern-Weil theory tells us that the Chern classes of a flat bundle over a manifold are always trivial in rational cohomology. But quite often they are non-trivial in integral cohomology, and hence provide a method of distinguishing between flat bundles. For instance, over a non-orientable surface, there are precisely two isomorphism types of flat vector bundles in each dimension (one being the trivial bundle), distinguished by their first Chern class in $H^2 (S; \mathbb{Z}) = \mathbb{Z}/2$.