Difference is that you can be androgyne without identifying yourself as bigender. You do not necessarily choose to be androgyne (either because of your genetics, or because the look you decided to have makes you look androgyne). It is just about appearance.

androgyne is a kind of presentation like feminine and masculine, an androgynous presenting person can’t be clearly read as male or female by binary thinking folks.

bigender is when you have two genders.

Also reminder that gender and presentation dont have to overlap (there are trans masc femboys).