Etale site is useful - examples of using the small fppf site?

Briefly, to understand $p$-phenomena in characteristic $p$ you need to replace the etale site by the fppf site. For example, to understand the $p$-torsion in the Brauer group you need the $p$-Kummer sequence and the cohomology of $\mu_{p^n}$, and the study of the $p$-torsion in the Tate-Shafarevich group entails the study of the cohomology of finite group schemes of $p$-power order over curves. There are also many purely geometric applications, e.g., to the Picard functor. You should think of the fppf cohomology as being THE cohomology theory, but the etale topology is fine enough for computing the cohomology of smooth group schemes, and the Zariski topology is fine enough for computing the cohomology of coherent sheaves.

The cohomology of $\mu_{p^n}$ and $\mathscr{A}[p^n]$ for an Abelian scheme $\mathscr{A}$ in characteristic $p > 0$ can also be studied using syntomic cohomology.