It seems to be an often repeated, "folklore-ish" statement, that diffeomorphism is an equivalence relation on smooth manifolds, and two smooth manifolds that are diffeomorphic are indistinguishable in terms of their smooth atlases.

There is a strange counter example in Lee's Introduction to Smooth Manifolds though, let us define two smooth manifolds modelled on the real line.

Let $\mathcal A$ be a smooth maximal atlas on $\mathbb{R}$ that is generated by the global chart $\varphi:\mathbb{R}\rightarrow\mathbb{R}$, $\varphi(x)=x$, and let $\bar{\mathcal A}$ be the maximal smooth atlas on $\mathbb{R}$ generated by the global chart $\bar{\varphi}(x)=x^3$.

The transition function $\varphi\circ\bar{\varphi}^{-1}$ is not smooth, so these two smooth structures are incompatible.

However the map $F:(\mathbb{R},\mathcal A)\rightarrow(\mathbb{R},\bar{\mathcal{A}})$ given by $F(x)=x^{1/3}$ is a diffeo, because $$ (\bar{\varphi}\circ F\circ \varphi^{-1})(x)=x, $$ and this map is smooth.

So the smooth manifolds $(\mathbb{R},\mathcal A)$ and $(\mathbb{R},\bar{\mathcal{A}})$ are diffeomorphic. Yet the two manifolds have incompatible, thus, different smooth structures.

**Question:** I guess I don't have a clear question, I am just somewhat confused. Because this is a *counterexample*, it seems to *prove* that the statement "two diffeomorphic manifolds cannot be told apart by their smooth structures" is wrong.

However **how** wrong it is? Can we consider the two manifolds given in this example equivalent? Is there any practical difference between the two? Is differential geometry the same on them?