"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 It sounds a bit outdated to me though.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)