You would basically never use an axe in woodworking. It's widely considered bad practice, produces unstable results, and there are better tools available
Well, I mean you could use an adze or a froe. Axes are really common in green woodworking (chair making) and they’re great for getting wood to split along the grain.
I second the green woodworking comment. An axe (and wedges) are a good way to hand-split logs into boards. And easier than doing it with a saw in some cases. Hand tool woodworking really makes you appreciate the multitude of weird old tools you find in antique stores and how clever people were when using human power before power tools.