A thorough analysis of what happened, both from inside-out and outside-in perspectives is in Peter E. Gordon, Continental Divide: Heidegger, Cassirer, Davos (Cambridge: Harvard University Press, 2012). The Davos Debate is seen as the point of origin responsible for many of the conflicts and alliances that led to the development of continental philosophy. That is why “Continental Divide” appropriately marks that nature of its result. It was a drama-filled event between these towering giants symbolic of a highly-anticipated confrontation between the old and new guard. Two years after Cassirer died in 1945, Eric Voegelin wrote a review of his Myth of the State which you can read here:
As you will see it is short but sweet! “It is a melancholy task” to write this review according to Voegelin, because Cassirer represents “an important phase in modern philosophy, that is the neo-Kantian movement” and its “passing away.” I would claim that the paramount shudder that Levinas and so many others felt from Davos and the dominance Heideggerian philosophy assumed thereafter speaks to groundbreaking, yet eclipsing shift in Western philosophy. Like Voegelin says about Cassirer’s death, and I believe the same can be said of his philosophical fate post-Davos, it “is rather the mark of a passing age than of the author.” The Davos encounter brought about the rupture of a new epoch where Cassirer’s humanism is called into question as an intellectual dead-end. It symbolizes the triumph of the post-humanist age. This is partially why Levinas could not ignore Heidegger, as can no one, despite his emphasis on ethics and our obligations to the “other.”
To the younger students, Cassirer represented the neo-Kantian relic despite any evidence to the contrary, whereas Heidegger was like a cultish, philosophical bad-boy who refused to take off his riding gear while Cassirer lectured. Heidegger, as Cassirer stated in one of his Davos responses, can also be labeled a neo-Kantian, although not of the Marburg school of Hermann Cohen’s tradition as was Cassirer. We are all philosophical cousins of Kant’s and the various philosophical schools throughout Germany subscribe to a unique interpretation of Kant’s life and work. For Heidegger and the Nazis, Cassirer comes from the idealistic-internationalist reading of Kant; what is called most harshly the Jewification of Kant. It is the camp which is known for its emphasis on logic, epistemology, and science. Heidegger’s Kant, on the other hand, lays the foundation for “fundamental ontology” and is a forerunner of phenomenology. Heidegger studied under Heinrich Rickert in Freiburg. When he moved in the direction of Husserlian phenomenology and then attempted to establish an existential ontology this led him farther away from the neo-Kantian movements. This certainly added to the characterization of Heidegger as a kind of philosophical maverick or renegade. Much of the motive behind Heidegger’s interpretation of Kant and others is to clearly distinguish his project from these schools. The crux of the matter partially lies with the historical embeddedness of Kant which is why the exchange at Davos was published in one of the appendices to Heidegger’s Kant and the Problem of Metaphysics (1997, 5th Ed.). So the essential confrontation is about who has Kant on their side or who can take possession of Kant and continue in that great tradition. It is quite clear that Heidegger won the day, but Cassirer’s work is undergoing a resurgence currently which I think his work is rightly due. Thanks for asking the question and there is no doubt that it was a pivotal moment in the history of Western thought.