One point compactifications of diffeomorphic $X$ and $Y$ I know that if two topological spaces $X$ and $Y$ are homeomorphic then so are their one point compactifications $X^*$ and $Y^*$. If $X$ and $Y$ (say both are smooth manifolds) are diffeomorphic what do we know about $X^*$ and $Y^*$? Are  they diffeomorphic too or only homeomorphic? If the latter is true, are there further condition under which $X^*$ and $Y^*$ will be diffeomorphic?

One point compactification of a manifold does not always have to be a manifold. For more details read this:
One-point compactification of manifold
But even when it is, I don't think it has to be unique. Please correct me if I'm wrong (manifolds is not my field of study).
For example take $M$ to be any exotic sphere of dimension $n\neq 4$. Then when you remove a point you obtain $\mathbb{R}^n$ as a topological space but since $n\neq 4$ then it is $\mathbb{R}^n$ as a manifold (the only exotic Euclidean space is $\mathbb{R}^4$).
Reversing, you have standard $\mathbb{R}^n$ which has at least two, non-diffeomorphic compactifications: the standard and exotic sphere.
