In French the translation is the same but I wonder why in English you have two words for this ? I think that's because there is a slight difference between these two words but I don't get what it is !
You now find a lot of people using admixture as a synonym of mixture; it is so widespread that this cannot be regarded as an error.
But the classic use of admixture is narrower: the prefix ad- (Latin = “to, toward”) implies that a new component or ingredient is brought “to” an original substance and mixed “into” it. We might for instance speak of the classic Manhattan cocktail as
rye whiskey with an admixture of vermouth and bitters.