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.
Best Answer
Pontryagin's original definition for his classes was an obstruction cycle as follows:
On the $n$ dimensional manifold $M$ take $(n-2i) +2$ vector fields in general position, and consider the points $x$ where they span a subspace (in $T_xM$) of dimension less or equal to $n-2i$. The set of such points $x$ form a cycle of codimenion $4i$ in $M.$ The dual cohomology class is $p_i(M).$
This definition might differ from the today accepted definition through Chern classes (as in the book by Milnor-Stasheff) by a second order class.