Defintion. An atlas is a set of pairs $\{(O_i,ψ_i)\}$ such that $\cup_iO_i=M$ and $\phi:O_i \rightarrow \mathbb{R}^n$, and most importantly, for $O_i \cap O_j \neq \emptyset $, $\tau_{ij}: \phi_i(O_i \cap O_j) \rightarrow \phi_j (O_i \cap O_j)$ is $C^{\infty}$.
However, Mathworld adds the following comment:
A smooth structure is used to define differentiability for real-valued functions on a manifold.
How does an atlas allow for one to determine which functions are differentiable and which are not?
If you have a function $f:O_i\cap O_j\to \Bbb R$, then $\phi_i$ and $\phi_j$ both make $f$ into a function on an open subset of $\Bbb R^n$ (say $f_i$ and $f_j$). It most certainly won't be the same function, but the fact that $\tau_{ij}$ (and $\tau_{ji}$) is $C^{\infty}$ means that $f_i$ is differentiable (or smooth) if and only if $f_j$ is.
That means that being differentiable (or smooth) with respect to some smooth atlas is a well-defined property; we say that a function $f$ on the manifold is smooth / differentiable whenever $f|_{O_i}\circ \phi_i^{-1}$ is for all $i$. However, a different atlas might give entirely different sets of differentiable / smooth functions.