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I don't think that $p_1$ distingushes the tangent bundles of exotic sheres $4$-spheres (if any). On an oriented smooth $4$-manifold $M$ Hirzebruch signature formula states that

$${\rm sign}(M)=\frac{1}{3}\int_M p_1(TM).$$

The signature of any homology $4$-sphere is zero since there is no homology in the $4$-th dimension.

1. There is one stupid way in which $p_1$ describes an obstruction, because $p_1$ is the $2$-nd Chern class of the complexification, and Chern classes have obstruction-theoretic descriptions.

2. The first Pontryagin class of a $4$-manifold $M$ appears in a nice integral formula of MacPherson and it involves the singularities of generic maps $M\to \mathbb{R}^4$. (I do not remember the reference at this moment.)

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1. I don't think that $p_1$ distingushes the tangent bundles of exotic sheres (if any). On an oriented smooth $4$-manifold $M$ Hirzebruch signature formula states that

$${\rm sign}(M)=\frac{1}{3}\int_M p_1(TM).$$

The signature of any homology $4$-sphere is zero since there is no homology in the $4$-th dimension.

1. There is one stupid way in which $p_1$ describes an obstruction, because $p_1$ is the $2$-nd Chern class of the complexification, and Chern classes have obstruction-theoretic descriptions.

2. The first Pontryagin class of a $4$-manifold $M$ appears in a nice integral formula of MacPherson and it involves the singularities of generic maps $M\to \mathbb{R}^4$. (I do not remember the reference at this moment.)