A Graphic Designer is a communicator first, artist second. A Digital Artist is an artist first, a communicator second.

In professional terms, a Digital Artist is someone that creates art via digital media. While all art is at it's essence communication, the term Digital Artist covers a very wide range of people and is less a title than a description of how they achieve what they're doing. Examples are: Concept Artists, Matte Painters, Storyboard Artists, 3D Artists, and some of the fine art fields like Painting, and Illustration (though obviously their work would be entirely digital).

A Graphic Designer is a visual problem solver; their job is to creatively solve various communication challenges via the combination of typography, graphics, imagery, and in some fields, sound (e.g. motion and web). Designers are facilitating a visual conversation between a business and their consumers. Whether that's a magazine, an online company, or a broadcast company, their job is to visually connect the two. Some examples of Graphic Designers are: Web Designers, Print Designers, Packaging Designers, Motion Graphic Designers, UI Designers, etc.

The biggest difference is in what they're communicating, not how they achieve it. The part that makes the distinction difficult is that the two fields often intersect. If I use a digital artists painting in a poster to advertise a movie, is it Graphic Design or Digital Art? I'd say the end product and it's intention make it Graphic Design, but it's Digital Art at the same time. As murky as the distinction can be in this situation, using the Digital Artist's work doesn't make me a Digital Artist anymore than it makes the artist a Graphic Designer.

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