Art is for artists. Design is not.
In reality, fine art and graphic design are not distinguished by commercialism but by objectives. The artist creates purely for personal expression. The designer creates to communicate a specific message to a specific audience.
Sometimes designers select their messages and audiences, but usually they do not. In any case, artists are free to express themselves as they like. Their works may communicate any message and evoke any and every emotion. But a designer must communicate a select message to a select audience, or their work is a failure.
Now, a designer may use art or even personal expression to convey a given message, but the primary goal is to communicate a unique message to a unique audience. This puts inherent restrictions on the works created by the designer.
Design is about problem solving. The problem is taking a unique concept and communicating it through as simple a means as possible so the concept is understood as easily as possible. This is not to say a particular message may not call for confusing, confounding, or angering an audience. But most will not. While art is often expected to be provocative or evocative, design usually communicates, and those communications are typically messages of informative, educational, or inspirational character.
Sadly, many designers don't see their role as communicative. They see websites (or videos, or print advertisements, etc.) as creative outlets and experimental spaces. They see their role as mostly creative and rarely consider accessibility. For them, function follows form.
The result is website designs that showcase the creative whims of egocentric designers instead of making content available for as many people as possible.
Let me be clear: this definition of art vs. design is my own. Most dictionaries--and many readers--are likely to disagree. But regardless of what terms you use, I believe the broad roles I describe are accurate distinctions. And I think it's crucial for every artist--or designer--to understand their role and the underlying motivations.