It's a proverbial phrase, which is best translated by a proverb that exists in English that means the same thing. In French it's bearskins, in English it's chickens. One of the joys of learning a new language!
They mean different thing. "Don't put the cart before the horse" is about doing/putting things in the right order, whereas "don't count your chickens before they hatch" is about not making unwarranted assumptions.