Jacket referred to a suit jacket, while coat referred to an overcoat or great coat.
The adjectives have been dropped from each, in most instances. It remains customary, for men's wear and ladies fashion, for a jacket to refer to a shorter length garment than a coat. So one would wear a winter jacket, which might be slightly longer than hip length, whereas as a rain coat or winter coat would be full coverage, closer in length to one's knees.
As far as material, there is nothing uniquely associated with either. "Leather jacket" is more typical, but that is because of the origin as motorcycle jackets, where leather is recommended for safety purposes. Yet I have seen the description "leather coat" for women's wear. Even though "wool coat" is more common, there are "wool jackets" too.
Then there is sailor's garb, known interchangeably as a pea coat or pea jacket. Modern usage of the words "coat" and "jacket" is fluid, not uniquely wrong or right, but instead, context-dependent.