There are classical existence results of a smooth structure on a topological manifold, and many results on the existence of multiple (i.e. exotic) smooth structures. Some utilize Freedman's theorem in dimension 4, and there's Milnor's twisted spheres in dimension 7, etc..
Morse theory and Riemannian geometry, for example, help to probe topology and only need a smooth structure. I questioned why we would want to know whether there is more than one smooth structure, and it leads to a less subjective question: Are there ways to study the underlying topological manifold from the existence of multiple smooth structures?
Here is a more refined question: Can different smooth structures give you different "bounds" on some topological quantities (maybe minimal genus of surfaces, "optimal" presentations of the fundamental group, etc.), or where one smooth structure gives you information while the other smooth structure can't give you any?