Stable differential structure It is known that $\Bbb{R}^n$ has a unique smooth structure unless $n=4$, which leads to an interesting consequence: the exoticness of non-stantard smooth structures on $\Bbb{R}^4$ can be eliminated by a smooth deformation with the help of an additional dimension due to the uniqueness of smooth structures on $\Bbb{R}^5$. In other words, the different smooth structures on $\Bbb{R}^4$ are not stably non-diffeomorphic.
Do there exist manifolds that admit stably non-diffeomorphic differential structures? To describe it precisely, two differential structures $M_1$ and $M_2$ on the same manifold $M$ are said to be stably non-diffeomorphic if for any smooth manifold $N$, $M_1 \times N$ is not diffeomorphic to $M_2 \times N$.
I'd like to know the different answers (if there really are) when $M,N$ are required to be boundless/closed.
 A: (Strictly speaking the answer is no since you allow $N = \varnothing$, so let me assume that $N$ should be non-empty.)
The following is not a complete answer since your notion of "stably non-diffeomorphic" is non-standard; usually one only allows products with $\mathbb{R}^n$. However, it may still be of interest to you.
Firstly, there is the product-structure theorem of Kirby-Siebenmann (Essay I.1 of Foundational essays...): it says that for a topological manifold $M$ taking the product with $\mathbb{R}^n$ with its usual smooth structure gives a bijection from the set of isotopy classes of smooth structures on $M$ to the set of isotopy classes of smooth structures on $M \times \mathbb{R}^n$, at least if $\dim(M) \geq 5$.
Secondly, a sufficient condition for two not-necessarily-homeomorphic closed manifolds $M_1$ and $M_2$ to not be stably non-diffeomorphic in your sense, is that there exists a homotopy equivalence $f \colon M_1 \to M_2$ such that $f^* TM_2 \oplus \epsilon^k \cong TM_1 \oplus \epsilon^k$ for some $k \geq 0$: there is a result of Mazur which says such an $f$ exists if and only if $M_1 \times \mathbb{R}^n$ is diffeomorphic to $M_2 \times \mathbb{R}^n$ for some $n \geq 0$. If you require all $N$'s to be closed, there is the following variant: if there exists a simple homotopy equivalence $f \colon M_1 \to M_2$ such that $f^* TM_2 \oplus \epsilon^k \cong TM_1 \oplus \epsilon^k$ for some $k \geq 0$, then  $M_1 \times S^n$ is diffeomorphic to $M_2 \times S^n$ for some $n \geq 0$. In fact, Mazur proved this with $D^n$ in place of $S^n$, but one can just take the double.
These two results are related as follows: by smoothing theory (Essay IV of Foundational essays...) the set of isotopy classes of smooth structures on $M$ with $\dim(M) \geq 5$ is in bijection with the set of homotopy classes of stable vector bundle structures on the stable tangent microbundle of $M$. Asking that two smooth structures $M_1$ and $M_2$ on a topological manifold $M$ have equivalent such stable vector bundle structures is asking whether for $f = \mathrm{id}_M$ it is true that $f^* TM_2 \oplus \epsilon^k \cong TM_1 \oplus \epsilon^k$ for some $k \geq 0$.
