Uncountable differential structures on $4$-manifolds?

The $1$-dimensional case is trivial, and the $2$-dimensional case is classical (but harder than one might expect). The case of dimension $3$ was proved by Moise in the 50s. Higher dimensions are different: The first distinct smooth structures on the same manifold were presented by Milnor for $S^7$. Furthermore, any compact, PL manifold in dimension $n\not = 4$ has only finitely many distinct smooth structures.

The case $n = 4$ is very different from the lower- and higher-dimensional cases, and the intuition there probably doesn't apply. The glib one-line answer is that in low-dimensions, geometry dominates; in high-dimensions, the $h$-cobordism theorem and its extensions dominate, and the subject becomes surgery theory. The problem with dimension $4$ is that the Whitney trick fails spectacularly. There are nice $4$-manifolds that have no smooth structure (i.e., a manifold $X$ not homeomorphic to any smooth manifold $Y$), and there are nice $4$-manifolds that have multiple smooth structures. For example, there exist uncountably many manifolds that are homeomorphic to $\mathbb{R}^4$, but no two of which are diffeomorphic. The $E_8$-manifold is compact and simply connected, but it can't be given a smooth structure. The details of which manifolds have a smooth structure are complicated, but the existence of a PL structure for a compact manifold $X$ is detected by the Kirby-Siebenmann class $\kappa\in H^4(X, \mathbb{Z}_2)$. In particular, if $X$ has dimension $<4$, then this class vanishes. (Of course, that's obscures exactly where the class comes from; it's a bit like reading the punchline without the joke.)

You asked for the intuition behind that, and the best answer I can come up with is that the naive intuition that one can take a improve a reasonable homeomorphism $X \to Y$ to a "nearby" smooth map fails completely in higher dimensions. Along similar lines, there are characteristic classes attached to manifolds that are invariant under diffeomorphisms but not under arbitrary homeomorphisms, and so the two categories of structures are distinguishable. Dimension $4$ is just particularly weird. Low-dimensional topology has a very different flavor from high-dimensional topology, and dimension $4$ is very different even from dimension $3$.

I would personally put it as follows:

Let $M$ be a differential manifold $M$ of dimension $\dim(M) < 4$ (this comes with a particular atlas), if we have $N$ a differential manifold of the same dimension and $M$ is homeomorphic to $N$ (so topologically the same), the $M$ and $N$ are even diffeomorphic.

Also, there are uncountably many $M_\alpha$, $\alpha \in A$, some uncountable index set, such that every $M_\alpha$ is a 4-dimensional differential manifold (including atlas etc.) such that for $\alpha \neq \beta$, we have $M_\alpha$ is homeomorphic to $M_\beta$ but $M_\alpha$ is not diffeomorphic to $M_\beta$.

For technical info, also see wikipedia, as usual.