“What is the difference between gesturing and gesticulating?

Gesticulating means movement without control.
The Art of Gesture, on the contrary, implies great control of your own gestures, total awareness of each limb, from hands to chest, to feet, all with great economy and harmony.
“Everything in nature has a language: people’s way of gesturing, gesticulating, their way of walking, of sitting down, of shaking hands… their way of using their voice and articulating their words… everything is an encyclopedia of invaluable signs.”

Dario Fo from “My First Seven Years (and a few more)”