I finally talked to Rob and did some literature search. Here are some examples of open subsets of Euclidean spaces which are homeomorphic but not diffeomorphic.

Let $\Sigma$ be an exotic $(n-1)$-dimensional sphere which can be realized as a Brieskorn variety (see here):
$$
V(a):=\{z\in S_\epsilon^{2m+1} \, : \, z_0^{a_0} + \dots + z_m^{a_n} =0 \},$$
where $2m=n$ and $S_\epsilon^{2m+1}=\{z: ||z||=\epsilon\}$ for some sufficiently small $\epsilon>0$.

For instance, any 7-dimensional exotic sphere will do the job.

Next, if $M^{n-1}$ is an $(n-1)$-dimensional (smooth) Brieskorn variety $V(a)$, it has a natural embedding in $S^{n+1}$ and the image has trivial normal bundle (since it is the regular level set of a smooth map to ${\mathbb C}$; this smooth map comes from the defining equation of $V(a)$). Hence, our $\Sigma$ embeds in $R^{n+1}$ such that the tubular neighborhood $U_\Sigma$ of the image is diffeomorphic to $\Sigma\times R^2$. Let $S$ be the standard $n-1$-dimensional sphere; it also embeds in $R^{n+1}$ with trivial normal bundle, of course, and we get a diffeomorphism $U_S\to S\times R^2$ for the tubular neighborhood $U_S$ of $S$ in $R^{n+1}$. It is then clear that $U_\Sigma$ is homeomorphic to $U_S$.

**Claim.** The open subsets $U_\Sigma$ and $U_S$ of $R^{n+1}$ are not diffeomorphic; equivalently, the manifolds $\Sigma\times R^2$ and $S\times R^2$ are not diffeomorphic.

This is where the story gets a bit complicated. Rob just says that he knows this, but does not remember a reference. The claim about nonexistence of a diffeomorphism appears on page 150 (Theorem 1) in

S. Kwasik, R. Schultz, Multiplicative stabilization and transformation groups. Current trends in transformation groups, (2002) 147–165.

But instead of a proof they provide six references none of which contains this claim (at least in a recognizable form). I have no reason to doubt that the proof can be extracted from some combination of their references. Instead, I looked in their other paper:

S. Kwasik, R. Schultz, Toral and exponential stabilization for homotopy spherical spaceforms. Math. Proc. Cambridge Philos. Soc. 137 (2004), no. 3, 571–593.

In that paper they pretty much provide a proof of this claim in section 4. (Their argument is used for space-forms.)

Here is a sketch of their argument. Suppose that there is a diffeomorphism
$$
f: \Sigma\times R^2\to S\times R^2.
$$
Let $S_a^1\subset R^2, S_b^1\subset R^2$ be concentric circles of radii $a$ and $b$, where $a$ is much larger than $b$. Then the submanifolds $S\times S^1_b$ and $f(\Sigma\times S_a^1)$ cofound a smooth compact submanifold $W\subset S\times R^2$. Next, one verifies that $W$ is an h-cobordism; since $\pi_1(S\times S^1)\cong {\mathbb Z}$, this h-cobordism is actually an s-cobordism and, hence, by the s-cobordism theorem, is diffeomorphic to the product. In particular, $\Sigma\times S^1$ is diffeomorphic to $S\times S^1$. Now, one can either refer to Theorem 1.9 in

R. Schultz, Smooth Structures on $S^p\times S^q$, Annals of Mathematics, Second Series, Vol. 90 (1969), p. 187-198

or recycle the above argument, to conclude that $\Sigma$ is diffeomorphic to $S$. This proves the claim.