For instance, consider the only concrete example in the article: the space of all possible configurations of a double pendulum is a manifold. The author claims it's useful to see it in a manifold, but why? Precisely, why more as a manifold than as a square [O,2π[²?
I also expected more talk about atlases. In simple cases, it's easy to think of a surface as a deformation of a flat shape, so a natural idea is to think of having a map from the plan to the surface. But, even for a simple sphere, most surfaces can't map to a single flat part of the plan, and you need several maps. But how do you handle the parts where the maps overlap? What Riemmann did was defining properties on this relationship between manifold points and maps (which can be countless).
BTW, I know just enough about relativity to deny that "space-time [is] a four-dimensional manifold", at least a Riemmannian manifold. IIRC, the usual term is Minkowski-spacetime.