I am looking for a cool and simple (but not worn-out) application of Bochner--Weitzenböck type formulas which could be used as a motivation.

>What is your favourite example?

**P.S.** Here is one which I like, but want more:
Assume two discs $\Delta_1$ and $\Delta_2$ with common boundary $\gamma$ bound a convex set in a positively curved three-dimensional manifold $M$. 
Then 
$\int_{\Delta_1}k_1\cdot k_2$
is small if the maximal angle between the discs on $\gamma$ is small;
here $k_i$ denote the principle curvatures of $\Delta_1$.