Stippled drawings are made up of tiny dots and are labor intensive. As with stippling, the dots are all the same size, and their closeness conveys the appropriate degree of darkness. The dots in standard halftones are not random at all but oriented on a grid. Stochastic images take better advantage of any printing press's capacities and so produce finer, more artful results than standard halftones ever can. The dots can be as small and as close as a particular printing device can print, rather than a multiple of that smallest dot as with standard halftones.
Resolution is higher with stochastic halftones.
Some color laser printers print stochastic halftones. I am particularly interested in lower resolutions for standard publication. I create my own stochastic images which printers treat as line art. Of course, the computer screen doesn't suffer from the same constraints as the printing press. On a printed sheet all the red ink must be the same red ink, not a lighter shade here and a darker one there, and the same for all the blue ink, and the yellow, and the black.