As a Western student who has a strong affinity for both, it seems to me that the differences seem to be a microcosm of the differences between more traditional Buddhism as the modern interpretation of it, which does tend strongly toward monism and pyrrhonism. For a long time, it seemed to me that claims of atman being universal was another method of arriving at the same conclusion as anatta, that is, the dissolution of ego boundaries.

Dzogchen as I understand it very strongly emphasizes the ground/rigpa, and I hope I am not overstepping my bounds by suggesting that an Advaitin would applaud this. The core doctrinal difference seems to be that Advaitins generally posit that something is, that is, Brahman, possibly even with qualities, whereas Dzogchen practitioners generally are more in the business of negating this.

Longchenpa's Treasury is perhaps the most astonishing set of texts on the subject of this thing, whatever you call it, to ever be translated into English. If the word rigpa were translated to Brahman instead, it would perhaps be the greatest Advaitic text in history, heretical as some of his technical assertions would have been. Despite doctrinal differences, I am quite convinced they come out of the same basic experience.