As far as I know, the terms sound designer associated to a precise job came up first in the movie industry on Apocalypse Now (1972 by Francis F. Coppola), for Walter Murch's remarquable work on that film.
After that day, he is the person who is responsible of every audio aspect of a project. He or she has a strong position and takes an important part of the creative audio crew.
His role can be compared with the role of sound supervisor or supervising sound editor. The first is the creative guy, the second is the technical/management guy.
Some are both depending on the movie.
But today this job is not only bound to the movie industry, indeed, TV, radio, video games, interactive web design, museography, even phone applications require a lot of sound, particular sound textures etc... so they are highly demanded.
Just for pleasure, you may also be interested in this :
In adverts, objectively, what could I say ?
Sound is as much important as image, sometimes more...