Functionally an armoire is a wardrobe. The distinction is one of style and interior fitting rather than purpose: armoires tend to be older or antique-styled, and more often carry shelves and drawers than a full-height rail.
Every armoire is a wardrobe. Not every wardrobe is an armoire.
Practical note: the shorter hanging space means armoires suit folded knitwear and shirts better than they suit coats and long dresses.