There's no important distinction between the two. And of course, Crewdson, Wall, Meyerowitz et al are actually highly commercial....they're seriously into peddling their names, just as Ansel Adams was. Ansel of course did a project for Kodak and a very long, highly advertised project for Polaroid...his Polaroids weren't wonderful but he did promote that company in his books...his books are the way many of his check-writing enthusiasts learned about him. Edward Weston became "hand to mouth" of course but he began first as a successful portrait photographer (like Imogene Cunningham) before he became poor. Nonetheless, it helped that Weston's collectors were the most wealthy people in California (and he made their portraits...see Daybook 2)...and it didn't hurt that some of his Mexico Daybook subjects were notrious/famous. And his son might not have gotten his perhaps-fame if his father wasn't so artistically successful.
The distinction between "art" photography and "commercial" photography is mostly that the artiste types anxiously sought and clung to employment at mediocre colleges whereas the best commercial types were selected by advertising agency art directors, who themselves were often artists....they knew what they wanted and knew how to get it.