I suppose we should start with defining the difference between a designer and art director.
In the Mad Men era of advertising, art directors would direct designers that put art boards together (mechanical assembly artists), illustrators,type houses, photographers/photo shoots and stock photography selects, in addition to commercials shoots, etc.
Typically, art directors concept based on strategy, they bring ideas through and connect the art with emotion. Designers are technical executors of the designs--from colors to typography. A List Apart has a nice article on it, and I recommend you read it. The chart below is from that ALA article.
But, copywriters really only have one lump name for the most part. They come up with concepts, use words to help bring an emotional or rational connection to the audience you're trying to reach.
Some of you might say, but what about technical copywriters? Well, they really aren't copywriters. They are writers who are write long form, detailed information such as instruction manuals, internally-circulated industry documents, assembly instructions, legal forms. I've also seen digital copywriter roles, which I'm not quite sure what that means (expect maybe that you understand digital or the role is at a digital agency).
And with digital, I've noticed that a lot of the way copywriter roles are treated is more akin to a designer role--execution only, whereas in traditional agencies, copywriter are much more used for their conceptual abilities, as well as crafting the words.
Now, I've seen many job listings for copywriters where it's more of an execution type of role, with little conceptual thinking involved. But, it's interesting that there is no distinction between the types of roles as there is for designers and art directors. Question is, should there be?