A principal bundle is <i> flat</i> if it has a flat connection, and the equivalence class of flat connections gives a <i> flat structure </i> on the bundle. Now of course, an isomorphism of principal bundles which admit flat structures, need not preserve the flat structures. For example, consider two homomorphisms $\pi_1(S)\to G$ that lie in the same path-component of the representation variety. The path joining them defines a flat bundle over $S\times I$, so by the covering homotopy theorem any two flat bundles in the path are isomorphic as principal bundles. On the other hand, two flat bundles are equivalent if and only if the corresponding homomorphisms are conjugate.

Exactly the same argument applies to non-principal flat bundles, e.g. bundles with fiber $F$ induced by homomorphisms $\pi_1(S)\to Diff(F)$.