An analytic structure on a manifold is an atlas such that all transition maps are real-analytic.
I am wondering firstly about times where a real manifold might arise where the full generality of a topological manifold is required, as opposed to a smooth or analytic one. Secondly I am wondering about times where it would be better to only consider the smooth structure.
Personally I find it something of a nuisance that not all topological manifolds can be triangulated. To me this is a sign of a definition which is unnecessarily general. But since I am no expert in this area, I am wondering what some uses might be or if people find reason to object.
Edit: I'd like to specify that I'm interested in examples of intrinsic interest where a topological manifold arose- not so much a distaste for topological manifolds.
Edit: It's normal to have opinions about proper mathematics, but as for this question about situations when purely topological manifolds arise, we would do well to ensure that these opinions not prevent the question from expression or distract from best-effort inquiry.
Best Answer
For $n \not= 4$, a smooth manifold homeomorphic to $\mathbf R^n$ is diffeomorphic to $\mathbf R^n$, so $\mathbf R^n$ as a topological manifold has just one smooth structure. The story is totally different for $n = 4$: look up exotic $\mathbf R^4$. There are $28$ different smooth structures on a $7$-dimensional sphere. That is, if you consider smooth manifolds that are homeomorphic to the smooth manifold $S^7$ but not necessarily diffeomorphic to it, there are $28$ examples. Look up exotic spheres.
In describing such phenomena, you could choose to talk about manifolds that are homeomorphic but not diffeomorphic or you could choose to talk about different smooth structures on a specific topological manifold. The choice is up to you.