The Schwarzian derivative (and 2nd derivative, etc.) are useful for 2
reasons:
- they allow you to apply calculus tools to discontinuous settings,
EG:
$f(x)=\begin{cases}x^4 & x\in \bQ & 0 x\nin\bQ\end{cases}$
This has a Schwarz 2nd derivative at $0$, even though it's only
differentiable at $0$.

- that is, they show that $n$th derivatives are -actually-
  infintesimal, even though we usually define them as the derivative
  of a derivative.
  (that is, you can take several derivatives at a point, even if you
  can't take the derivative anywhere else)

- it shows how $n$th derivative is convolution with a distribution
  ...and once you know about distributions, it's clear how
  (that is, the distribution is the limit of a combination of delta
  functions)
