I was thinking about this for a while. The definition I use for the smooth manifold is the same as per Wikipedia. Let $\{(U_k,\phi_k)\}$ is a smooth atlas of $M$. Then the natural atlas which was coming in my mind for $M-M$ was $\{(U_i - U_j, \phi_i - \phi_j)\}$ where $(\phi_i - \phi_j)(u)=\phi_i(u) - \phi_j(u) \; \forall u \; \in U_i - U_j$. Is my approach correct?
By $M - M$ I just mean formal difference of two sets where an element $x$ of $M-M$ can be written as $n-m$ for some $n,m \in M$. Note that "difference" in $M-M$ has no meaning but I am seeking a suitable atlas for this set so that when I am in some $\Bbb R^n$, I will do subtraction as per the addition is done in the group $\Bbb R^n$.
As a beginner in learning the subject, I am not confident in writing down the details. Thank you.
EDIT: After a recent comment by @MikeMiller, I realized that I was actually working in $M \times M$. So I thought to change my definition of $M - M$. I see now $M - M$ as a set of equivalence classes where the equivalence relation is such that any to pairs $(m,n)$ and $(p,q)$ (or $m-n$ and $p-q$) are equivalent if we have a suitable atlas for $M-M$ such that in local coordinates, $m-n=p-q \in \Bbb R^n$. The problem is I want to know whether such an atlas exists.
This is not an answer to your question but I feel it's too long to be included in a comment. I just want to mention that there's an interesting result regarding the Minkowski sum of $2$ convex sets with smooth (of class $C^\infty$) boundary.
Smoothness of Vector Sums of Plane Convex Sets
The main result is that the sum of $2$ convex sets with $C^\infty$ need not have $C^\infty$ boundary. In fact we only get the smoothness of the boundary up to class $C^{20/3}$. This might suggest that the answer to your question could be negative (thought I am not sure since your question right now is not well-defined).