Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal


$$
w(TM)=w(TN)?
$$ 
Do the Chern classes of the tangent bundles of $M$ and $N$ equal


$$
c(TM)=c(TN)?
$$ 



characteristic classes of homotopy equivalent manifolds
Wu's formula for the Stiefel-Whitney classes implies that they are invariants of homotopy type. See for example [here](http://www.map.mpim-bonn.mpg.de/Wu_class).


Chern classes are not even diffeomorphism invariant, and it is possible to have two complex structures on the *same* manifold with different Chern classes. See [this](https://mathoverflow.net/questions/70429/is-canonical-class-a-topological-invariant) question and its answers.


Also, Pontryagin classes are not invariants of homotopy type. As Danny mentions below, there are examples of fake projective spaces with different Pontryagin classes than $\Bbb C P^n$, but I don't have an explicit example. [This](http://projecteuclid.org/euclid.pjm/1102970912) paper describes how surgery theory can be used to start with a parallelizable manifold of a certain form and obtain a non-parallelizable manifold which is homotopy equivalent to the original. An interesting question is which rational polynomial expressions in the Pontryagin classes are homotopy invariants. The [Novikov conjecture](http://www.scholarpedia.org/article/Novikov_conjecture) asserts that only the $L$-polynomials (from the Hirzebruch signature theorem) are.


