I was wondering if any vector bundles on a manifold other than the tangent bundle give topological invariants. I guess stiefel Whitney classes also come from the inverse bundle  but other than that.
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The top exterior power of the tangent bundle determines orientablility (the most basic of topological invariants, after dimension) in the sense that $\wedge^nTM$ is a trivial line bundle iff $M^n$ is orientable. The complexified tangent bundle $TM\otimes_{\mathbf{R}} \mathbf{C}$ is used to define the Pontryagin classes, which are themselves smooth invariants (although the Pontryagin numbers and the rational Pontryagin classes turn out to be topological invariants). Both of these examples are built from the tangent bundle, so I suspect are not what you are looking for. Aside from this you could look at the stable normal bundle (as you suggest) or put some additional structure on your manifold (as Paul Siegel suggest in his comment). 

