[The title above was conceived as an initial answer to the question posed below about the main ways Hegel extends Aristotle, but the article then wanders away from Hegelian genealogy in further pursuit of that question.]
Hegel was at the same time deeply Aristotelian, deeply Kantian, and highly original. Across numerous posts, I have been pointing out Hegel’s connections with Aristotle. This implicitly poses the question, how should we summarize the aspects of Hegel’s contributions that go beyond Aristotle?
What Brandom has called Hegel’s genealogy captures most of this at a high level. A Hegelian genealogy is a recollective making more explicit of our current best self-understanding in terms of a backward projection of part of that current understanding onto what we take to be its historic roots, in order to then trace a sequence of its development into our full current understanding. I would note that this sort of understanding involves the kind of interweaving of history and creative fiction that has been discussed at length by Paul Ricoeur.
Hegel is at one with Aristotle in recognizing that the end goal of a process is emergent rather than pre-established from the beginning, as someone like Leibniz or Plotinus might suggest. He does not mean to literally assert, e.g., that Socrates already explicitly thought in terms of German Idealist concepts like Subjectivity and Freedom. In part, he is deliberately using anachronistic terms as a sort of pedagogy for a contemporary audience. More significantly, he is making a historical claim based on current understanding that the roots of German Idealism go all the way back to Socrates.
On the other hand, while Aristotle and Hegel are both very concerned with development and take a retrospective perspective on it, Aristotle does not explicitly address the development of large cultural formations or development over long periods as Hegel does. Aristotle takes large formations in a mostly synchronic way.
On a small scale, while Aristotle makes heavy use of both material incompatibility and material consequence, he does not tightly combine these as Hegel does.
Aristotle recognizes that a concern for error and its rectification is integral to the pursuit of truth, but does not apply this to whole social formations or historical periods the way Hegel does. He does not have Hegel’s positive vision of the necessity of error for learning, and of a path to greater rationality that can only be achieved through the successive resolution of errors.
Aristotle treats mutual recognition as an important part of the description of the key ethical goal of friendship or love. Hegel further develops the idea of mutual recognition, makes it more primary, broadens its applicability, and also uses it to explain how normativity is socially and historically constituted.
Hegel also takes over Kant’s idea that normativity forms an outer frame around all other concerns. (See also Aristotle and Kant; Brandom and Kant; History of Philosophy; Edifying Semantics; Mutual Recognition.)