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 $3$.
Yes, it is true. For $n>8$ there exist exotic $n$-spheres if $n$ is congruent modulo 192 to one of 2, 6, 8, 10, 14, 18, 20, 22, 26, 28, 32, 34, 40, 42, 46, 50, 52, 54, 58, 60, 66, 68,
70, 74, 80, 82, 90, 98, 100, 102, 104, 106, 110, 114, 116, 118, 122, 124, 128,
130, 136, 138, 142, 146, 148, 150, 154, 156, 162, 164, 170, 178, 186.
See Corollary 1.5 of "The 2-primary Hurewicz image of tmf" by Behrens, Mahowald, Quigley.
Best Answer
Actually, there is a (nontrivial) theorem somewhere in the vicinity of what you are trying to ask. The right objects to consider are triangulated manifolds (more precisely, PL triangulations, of manifolds, i.e. where links are PL spheres). (PL stands for "piecewise-linear," this is a generalization of the notion of a piecewise-linear function you might be familiar with.) Every triangulated manifold $M$ defines a graph (the 1-dimensional skeleton of the triangulation), but a triangulation actually contains much more information than that graph.
Every manifold admits infinitely many triangulations, provided that it admits one. Thus, the natural notion replacing diffeomorphism for smooth manifolds is the one of a PL homeomorphism. Equivalently, you can say that two triangulations are regarded as "the same" if they admit isomorphic subdivisions. Thus, one defines the notion of PL isomorphic triangulated manifolds. Now, one can ask:
Is there a (PL) triangulated manifold which is homeomorphic to ${\mathbb R}^4$ (equipped with the "standard" PL structure, obtained by a suitable subdivision of the standard cubulation) as a topological space but is not PL isomorphic to such ${\mathbb R}^4$.
The answer to this is indeed positive and the proof uses the result about the existence of exotic differentiable structures on ${\mathbb R}^4$. The proof boils down to nontrivial a theorem (due to Kirby and Siebenmann) that in dimensions $\le 6$ the categories PL and DIFF are naturally isomorphic. In particular, if $M, M'$ are smooth 4-dimensional manifolds which are homeomorphic but not diffeomorphic then they can be PL triangulated so that the resulting PL manifolds are not PL isomorphic. From this, it follows that for every exotic smooth ${\mathbb R}^4$ there exists an exotic PL triangulated ${\mathbb R}^4$.