This is a loose question (more philosophical than mathematical, I guess):
One definition reads: An exotic sphere is a manifold homeomorphic to the sphere, and not diffeomophic to it.
When you take this literally, these are just homeomorphic copies of the sphere, in which you cannot "smooth out all the kinks" (i.e. not diffeomorphic) in a way (it is never good too replace the mathematical definitions by "Kindergarten language", but from a distance, this is not far from the truth).
I have also read somewhere that an exotic sphere has a different smooth structure than your regular $(S^n, id)$. So my question is:
Consider an exotic copy $(\Sigma^n, \phi)$ of $(S^n, id)$. Is $(\Sigma^n, \phi)$ a smooth manifold by its own, or is it just a topological copy of it ?
The very first time an exotic sphere was built was by Milnor, and he defined them as total spaces of $\Bbb S^3$-bundles over $\Bbb S^4$. From this angle, it does not quite look like he started from $\Bbb S^7$ and did something to the smooth structure.
Historical considerations apart, I can see no reason not to consider an exotic sphere as a smooth manifold in its own right.