Let $M$ a manifold. Two atlas $A_1=\{(U_i,\varphi_i)\}$ and $A_2=\{(V_i,\psi_i)\}$. They define the same smooth structure if whenever $f$ is smooth wrt $A_1$ then it will be smooth wrt $A_2$. For example $$A_1=\{(\mathbb R^n,id_{\mathbb R^n})\}$$
and $$A_2=\{(B_1(x),Id_{B_1(x)})\mid x\in \mathbb R^n\}$$
are two different atlas but they define the same smooth structure.
Do you have an example (on $\mathbb R^n$ or any easy smooth manifold) that describe two different smooth structure ? I don't really see it.
Best Answer
Here are two different smooth structures on $\mathbb{R}$ : $\{(\mathbb{R},\operatorname{id})\}$ and $\{(\mathbb{R},x\mapsto x^3)\}$.
Using the same idea, you can prove that there exists an uncountable set of smooth structures on $\mathbb{R}$.
If $M$ is a topological manifold, its homeomorphisms group acts on its set of smooth structures by: $$f\cdot\{(U_i,\varphi_i)\}_{i\in I}=\{f^{-1}(U_i),\varphi_i\circ f\}_{i\in I},$$ and generally an orbit of a given smooth atlas is uncountable.