“Visage” is of more common use.

I guess the main difference is that “visage” refers expressly to the face as the front part of the head.

“Figure” have a more general meaning but it’s still a perfectly acceptable translation for face. It sounds a bit outdated to me though. My grandma used to tell me “lave toi la figure” > “wash your face”

Other uses of “figure” (you cannot swap it for “visage” here):

Une figure historique : a historically important person, a person that was so important for an event / movement … that it has come to embody it

Une figure géométrique : a geometrical shape like a circle, a square…

Faire bonne figure : to put on a brave face / avoid looking bad (even though you have reason to do so)

Yes visage is more formal. It is used for example in a beauty salon (soins du visage) or in a medical context (greffe du visage.) No way for “Soins de la figure” or “greffe de la figure” to be used here.

On the other hand, kids are much more likely to say “elle s’est peint la figure” than “elle s’est peint le visage” and I would say to my kids “lavez-vous la figure !” definitely not “lavez-vous le visage !“

There are also face, bouille, poire, faciès, gueule, portrait, trogne, trombine…