TITLE: What is geometrically the Pontryagin class?
QUESTION [42 upvotes]: What does the Pontryagin class detects or is an obstruction to? Please avoid any answer using that it's the even Chern class of the complexified bundle or any interpretation that relies on the complexified bundle. 
As related question might be the following: when one defines the obstruction classes on a rank $4$ vector bundle (and if the first three obstruction classes do vanish) then the fourth obstruction class can be decomposed as the Euler class and the first Pontryagin class (as $\pi_3(SO_4) \simeq \mathbb{Z} \oplus \mathbb{Z}$). Is there a geometric description of a system of generators in $\pi_3(SO_4)$ which is associated to these classes?
EDIT: deleted "For example, why does the first Pontryagin class distinguishes the (tangent bundles of the) exotic $4$-spheres?" as it is wrong, see Liviu's answer below.

REPLY [7 votes]: Worth mentioning separately I believe: in "A combinatorial formula for the Pontrjagin classes", Gelfand and MacPherson construct something like analogs of Segre classes for Pontryagin classes: for a triangulation of a manifold $X$ they invoke oriented matroids to produce explicit rational simplicial cycles on its barycentric subdivision which are Poincaré duals of inverses $\bar p_i(X)$ of the Pontryagin classes of $X$.
They also describe (on half a page!) a version of the Chern-Weil theory for Pontryagin classes of a vector bundle $E$ with connection on a manifold $M$ which shows relationship between their approach and the "standard" one. It is so concise and enlightening that I decided just to reproduce it here. They consider the Grassmanian bundle $\pi:\mathscr Y\to M$ of codimension 2 planes in $E$, together with the principal bundle $\rho:\mathscr Z\to\mathscr Y$ corresponding to the tautological quotient 2-plane bundle over $\mathscr Y$. The connection on $E$ gives them a 1-form $\Theta$ on $\mathscr Z$ with coefficients in the orientation sheaf of $\mathscr Z$ and a curvature form $\Omega$ on $\mathscr Y$ determined by $\rho^*\Omega=d\Theta$. Their formula then is$$\bar p_i(E)=(-1)^i\pi_*\Omega^{\dim(E)-2(i-1)}.$$