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Integer pontrjagin classes are diffeomorphism invariant, while rational pontrjagin classes are homeomorphism invariant, due to Novikov. Also there are examples where two smooth manifolds are homeomorphic but with different integer pontrjagin classes. And of course the cohomology of the manifolds need to have some torsion in order to make this work. See for example, Matthias kreck and Wolfgang Lueck's book, The novikov conjecture - geometry and algebra, pp 29- 31
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Integer pontrjagin classes are diffeomorphism invariant, while rational pontrjagin classes are homeomorphism invariant, due to Novikov. Also there are examples where two smooth manifolds are homeomorphic but with different integer pontrjagin classes. And of course the cohomology of the manifolds need to have some torsion in order to make this work.