Intuitively, I would like to say that the tangent bundle of $S^2$ and the bundle ($S^2\otimes \rm{y-z\ plane}$) is different. By the latter I mean a product bundle, embedding in $R^3$ it is equivalent to attaching a yz plane at each point of $S^2$.
Is there a type of characteristic class that tells these two apart? The Pontryagin class for both is $0$ as it is $2d$ by $2d$.