There is no difference.
Design is art, art is design.
Any good artist is designing their work. Any good designer has an art to their work.
The artist "designs" the composition of any piece they create, be it a painting, sculpture, furniture, etc. There is design even in a child's drawing. They choose where the sun goes, where the dog is, where the grass stops, etc. That's all design within artwork.
The designer uses "artistic" choices to create appealing images. Colors, line weight, position, proximity, scale, motion, are all artistic choices made by a "designer".
If any possible difference does exist, it could possibly be argued that there is a responsibility difference. But that does not convey to any of the work specifically. Those who self-classify themselves as "designers" may perhaps be more time clock punchers while creating art. And those who self-classify as an "artist" may be slightly more free to create their designs when the mood strikes them.
I see this question as being similar to asking, "Is that a dollar or a buck?", "A quid or a pound?" Same thing.... just different terminology.
Designer: a person who devises or executes designs, especially one who creates forms, structures, and patterns, as for works of art or machines.
1 a person who produces works in any of the arts that are primarily subject to aesthetic criteria.
a person who practices one of the fine arts, especially a painter or sculptor.
a person whose trade or profession requires a knowledge of design, drawing, painting, etc.: a commercial artist.
a person who works in one of the performing arts, as an actor, musician, or singer; a public performer: a mime artist; an artist of the dance.
a person whose work exhibits exceptional skill.
I've crossed out obvious unrelated definitions, but the other three are pretty much the same as the definition of "Designer"
In today's world, many may see a "designer" as a computer operator and sadly many designers may see themselves that way as well. However, design traditionally has required great craftsmanship, steady hands, a good artistic eye, etc. Design hasn't always been as simple as launching an application and them moving some objects around on a digital page. There is art in it, same as there is design in art.
I, personally, don't know any designers who would not also qualify themselves as artists. I do know several artists who would not qualify themselves as designers, but that's more a bigoted stance about money than the actual work.
Were the designers who created the wonderful music posters and album covers of the 60s and 70s not artists?
Was Paul Rand not an artist?
Saul Bass and Milton Glazer merely designers with no "art" in their work?
I think not.
Oh so many down votes.... :) Apparently a great deal of designers think they aren't artists or a great deal of artists think they aren't designers.