Recently I have been learning about cobordisms. While I have seen many applications of cobordism to stable homotopy theory and immersion problems, I couldn't find an example of spaces that can be distinguished by cobordisms as a (co-)homology theory that cannot be distinguished by ordinary homology and K-Theory.
Explicitly I would like to know an example for two CW-complexes X,Y Such that the algebras $MU^* (X)$ and $MU^* (Y)$ are not isomorphic, but $H^* (X,\mathbb{Z})\cong H^* (Y,\mathbb{Z}), K^* (X)\cong K^* (Y)$ and $H^* (X,\mathbb{Z}/p) \cong H^* (Y,\mathbb{Z}/p)$ as modules over the Steenrod-Algebra.