Intuition/meaning behind/physical content of the concept of a smooth structure Some mathematical structures are visualized very well. I imagine how a shapeless bunch of points (a set; the only property of which is quantity) is collected in one or another soft form (topological space), depending on what topology is set on it. I imagine how this soft (floating) shape becomes hard when I set the metric. The first examples of deformation retractions give a vivid geometric intuition of the homotopy type as a "(homotopy) framework". Thus, the concepts of homotopy type, topology, and metrics have a clear physical content. In contrast to these three examples, for the concept of a smooth structure on a manifold, everything is not so clear due to the fact that manifolds up to dimension 3 have a unique smooth structure

Do you have any intuition for a smooth structure on a manifold? Do you
see any meaning/physical content in it? Do visually homeomorphic
non-diffeomorphic spaces differ for you?

After a smooth manifold is provided with additional structures (Riemannian,  symplectic, etc.), the meaning of these objects becomes clear. But a smooth manifold devoid of additional structures is still very mysterious for me.
P.S. I'm not sure if this question is for a forum (on the other hand, the system showed me many questions with similar titles). If so, then feel free to close it.
 A: $\newcommand{\R}{\mathbb{R}}$The purpose of a manifold structure is to be able to differentiate functions. And initially we know how to differentiate functions only if the domain is an open set in $\R$. Moreover, once you know how to differentiate functions, this opens the door to doing all the other stuff you do with manifolds.
Here's how I think of it: Initially, think of a manifold as just a set $M$ and nothing else. Think of a coordinate map as a bijection $\phi: O \rightarrow \R^n$, where $O\subset M$ and $\phi(O)$ is an open subset of $\R^n$. An atlas is a collection of coordinate maps such that the domains of the coordinate maps cover $M$. No assumptions on topology or differentiability yet.
A topological manifold is $M$ with an atlas $\mathcal{A}$ such that for any two coordinate maps $\phi_1: O_1 \rightarrow \R^n$ and $\phi_2: O_2 \rightarrow \R^n$ such that $O_1\cap O_2\ne \emptyset$, then the map $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$ is a homeomorphism. Notice that such an atlas immediately defines a topology on $M$ where the domains of the coordinate maps in $\mathcal{A}$ form a base of open sets. Assume that this topology is Hausdorff, and you have a topological manifold.
A smooth manifold is defined in exactly the same way, except you assume that the change of coordinate maps, $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$, are smooth.
Clearly, a smooth manifold is a topological manifold. If you start with a topological manifold, extend its atlas to a maximal atlas, then you can ask whether there is a subatlas (i.e., a subset of coordinate maps) that satisfies the definition of an atlas of a smooth manifold. If so, you say that the topological manifold is smoothable.
There is no reason why there couldn't be two different smooth subatlases of a topological manifold. And, if there are two such subatlases, there is no reason why they should be compatible with each other. In other words, if you have a coordinate map $\phi_1: O_1 \rightarrow \R^n$ in the first smooth atlas and a coordinate map $\phi_2: O_2 \rightarrow \R^n$ in the second smooth atlas, it does not necessarily follow that $\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)$ is smooth. It is homeomorphic, since both lie in the topological atlas.
Now, to make things even more complicated, it is possible that there is a global homeomorphism $\Phi: (M,\mathcal{A}_1) \rightarrow (M,\mathcal{A}_2)$ that is a smooth diffeomorphism of the two apparently different smooth manifolds. So, even though the two subatlases are incompatible, they actually define two smooth structures on $M$ that are diffeomorphic.
Finally, we say that a topological manifold has more than one smooth structure on it if there are two smooth subatlases such that no map $\Phi$ as described in the previous paragraph exists.
