Matt Dinan of Holy Cross comments:
I think that the biggest difference between Hegel and Kojeve (or at least Kojeve’s Hegel) is that Hegel’s philosophy of absolute spirit doesn’t make a whole lot of sense without Christianity, whereas Kojeve thinks that Christianity can be altogether superseded. For the actual Hegel Geist’s process of historical unveiling is actually the revelation of the nature of the divine in history. Philosophy’s claim to absolute status in Hegel is predicated upon its ability to mediate the substantive truth of Christian revelation (FREEDOM) from an abstract into a concrete truth. Thus it’s not Nietzsche who first posits continuity between the infinite subjectivity of Socrates and Christianity; it’s Hegel (except Hegel sees this is entirely positive, not damning). As Stanley Rosen points out somewhere, Kojeve’s is a really radical and unorthodox reading of Hegel, and it’s probably not done Straussianism any favors as a movement to take up that reading of Hegel so uncritically.
There’s actually a new book on this very subject coming out soon.