With the risk of making a simple thing confusing, here's another point of view.

Two functions $f,g$ on an interval $I$ give a map to the projective line $I \dashrightarrow \mathbb P^1$ by $x\mapsto [f(x):g(x)]$ (some indeterminacies can occur if both functions vanish at some point).
You can ask: when does this map have a critical point?

If you choose the chart $[y:1]$ on $\mathbb P^1$ you have to look at the vanishing of the derivative of $f/g$.

If you lift the map to $I\to \mathbb R^2$ via $x\mapsto (f(x),g(x))$ then you get a curve in the plane, with tangent vector $(f'(x),g'(x))$. The map to $\mathbb P^1$ has a critical point precisely when the vectors $(f(x),g(x))$ and $(f'(x),g'(x))$ are proportional, so you get the Wronskian.

The book of Ovsienko and Tabachnikov "Projective Differential Geometry, Old and New" contains a wealth of additional information.