In applications of the fundamental group(oid) to smooth manifolds it is sometimes useful to have paths which are smooth, rather than merely continuous. For example, if we consider the local system of solutions of some linear partial differential equation, we can pull it back along a path, where it can be viewed as the solutions of an ordinary differential equation. The "nicer" the path is, the "nicer" will be the coefficients of the resulting ordinary differential equation.
It seems impossible to simply define this issue away by only allowing smooth paths in the first place, since a concatenation of smooth paths need not be smooth. Another possible problem: if we want to also restrict to smooth homotopies, we have to figure out what this means at the corners of the square.
Here's a couple of specific questions, since the previous paragraphs are so vague. Does every homotopy class of paths have a smooth representative? If two smooth paths are homotopic, can the homotopy be chosen to be smooth? How does one deal with the corners of the square? I imagine that the answers to these questions are written down somewhere.