I am not an expert but I guess there is a number of "historical" reasons explaining the lack of exploration of physical consequences of exotic differential structures:
1) many physicists are inclined to keep things as simple as possible;
2) the existence of exotic differential structures on $\mathbb{R}^4$ is just not sufficiently well known in the physics community (how many textbooks on differential geometry mention this?).
That said, a bit of googling has brought up (in addition to the book pointed out by Steve Huntsman in the comments) a 1989 paper Topological defects and differential structures by R. Rohm exploring some of the possible physical manifestations of exotic differential structures of spacetime, the 1994 paper Exotic Smoothness and Physics by C. Brans, the 1996 paper Exotic smoothness, noncommutative geometry, and particle physics by J. Sladkowski (preprint version is on arXiv), and, more recently, the preprints Exotic Smoothness and Quantum Gravity by Asselmeyer-Maluga , On the geometrization of matter by exotic smoothness by him and Helge Rosse, Exotic smooth $\mathbb{R}^4$, noncommutative algebras and quantization, Exotic smooth $\mathbb{R}^4$, geometry of string backgrounds and quantum D-branes and Gerbes, $SU(2)$ WZW models and exotic smooth $\mathbb{R}^4$other preprints by Asselmeyer-Maluga him and Królhis coauthors, and the papers Exotic Smoothness and Noncommutative Spaces. The Model-Theoretical Approach and Exotic Smooth 4-Manifolds and Gerbes as Geometry for Quantum Gravity by Król.
