They both went after the mass market with solid sets at a reasonable price. AK was out of business by the mid-1930s, while Crosley lasted much longer.
For the time they were in competition, I sense that Crosley had the lower price point, squeezing good performance out of low tube-counts and compact cabinets.
:Is there a connection between Atwater and Crosley. Some of their early products look very similar, especially the speakers. Crosley sounds more like Cmdr. Mcdonald of Zenith who hired radio experts to get the wheels rolling.