As I was learning about galilean space-time structures (through Arnold’s Mathematical Methods of Classical Mechanics), I found things to be helpful for me to understand the meaning of compatibility of smooth structures. Let $M$ be a set and $\psi_1 : M \to \mathbb{R} \times \mathbb{R}^3$ be any (as wild as you wish) one-to-one and onto map. The function $\psi_1$ is then said to be defining a Galilean coordinate system on $M$. And a coordinate system $\psi_2$ is said to be moving uniformly with respect to $\psi_1$ if the map $ \psi_2 \circ \psi_1^{-1} : \mathbb{R} \times \mathbb{R}^3 \to \mathbb{R} \times \mathbb{R}^3$ is a galilean transformation on $\mathbb{R} \times \mathbb{R}^3$; at which case, we say $\psi_1$ and $\psi_2$ give $M$ the same galilean structure.

A Galilean transformation is easily imaginable; it can be proved to be just a combination of the rotation of the inertial observer, and uniform movement of him/her with a fixed velocity vector, and an adjustment in our clock. These transformations are the ones that are irrelevant to the physical phenomena that is happening. That is to say, $\psi_1$ and $\psi_2$ are assigning the same physical structure to $M$ (time function is the same, the distance function for the simultaneous events is the same, etc.). An example of a galilean structure that is not the same as $\psi_1$ is a map that first applies $\psi_1$ itself, and then stretches the time coordinate by a factor of $2$; name this map $\psi_3$. To see that $\psi_3$ is not the same structure as $\psi_1$, consider the inverse image of the time axis of $\mathbb{R} \times \mathbb{R}^3$, i.e. $T = \psi_1^{-1} \big( \mathrm{span} (1, 0, \dots, 0) \big)$; also we have $T = \psi_3^{-1} \big( \mathrm{span} (1, 0, \dots, 0) \big)$. Moving along $T$ (which is the piece of $M$ that is meant to encode passage of time at the origin of space), and watching the images by $\psi_1$ and $\psi_3$, we can see that time passes with double-speed in $\psi_3(M)$ compared to $\psi_1(M)$. Therefore, the structures that $\psi_1$ and $\psi_3$ define on $M$ are different as space-time structures.

Same is the story in the theory of smooth manifolds. Say $X$ is a two-dimensional manifold and $p$ be a point in $X$. If $(U, \varphi_1)$ and $(V, \varphi_2)$ are charts about $p$, we call the two charts compatible if the map $\varphi_2 \circ \varphi_1^{-1}: \varphi_1 (U \cap V) \to \varphi_2 (U \cap V)$ is a diffeomorphism. We may confine our attention to $U \cap V$, because this is the place where talking about compatibility is more meaningful. Just like above, the maps $\varphi_1$ and $\varphi_2$ are assigning the smooth structure of $\mathbb{R}^2$ to $X$ in their own way. An analogue example of the incompatible structures above could be the following: Let $X = \mathbb{R}^2$ and suppose $L$ is the graph of the absolute value function residing in $X$. Suppose $\varphi_1$ is the identity map, and $\varphi_2(L)$ is a straight line in $\mathbb{R}^2$; then $L$ with the structure coming from $\varphi_1$ is considering $L$ as a non-smooth curve, but with the structure from $\varphi_2$, it is actually a smooth curve in $X$. Here, diffeomorphisms are picked to be the group of maps that mod out all the differences that are unimportant from a differentiable viewpoint (just like the way that galilean transformations group helped us dismiss rotations, clock changes and uniform motion of the observer). Therefore, the diffeomorphism group (along with the galilean transformation group in the above scenario) is defining insignificant differences; whatever difference that cannot be removed by applying a member of the group, is considered as a real difference between the structures. Like in our example, if $\gamma$ is any diffeomorphism of $\mathbb{R}^2$, then $\gamma \circ \varphi_1(L) = \gamma (L)$ will still have the edge-point and cannot be equal to a straight line, so $\varphi_1$ cannot become $\varphi_2$ by means of an unimportant change (i.e. a diffeomorphism like $\gamma$), hence these structures are different.