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Fonts, vector graphics, interpolation for animation and video games

I think that, nowadays, it’s more likely that fonts use quadratic Bézier curves (as in TrueType and its derivatives), rather than cubic ones (as in PostScript and Adobe’s Type 1 fonts), but since OpenType supports both (https://en.wikipedia.org/wiki/OpenType#Description) I may be wrong.

Similarly, SVG supports both quadratic and cubic Bézier curves.

And cars. Curves with higher-order continuity is why modern cars have such interesting smoothly curved surfaces.

This was the raison d'etre for Bézier curves. Its inventor and re-inventor worked at car companies.

https://en.wikipedia.org/wiki/B%C3%A9zier_curve#Invention

Roads and tracks. Connecting straightaways and turns to at least second order (sometimes third, esp. for high speed trains) makes transitions much smoother as it reduces or eliminates jerk.

https://en.m.wikipedia.org/wiki/Jerk_(physics)

Even during driving, intuitively you don't yank the steering wheel to the desired position but produce a eased-in eased-out progressive movement that results in such a trajectory.