It was no sophomoric error when Friedrich Engels described Aristotle — not Plato or some Neoplatonist — as the greatest dialectician of the ancient world.
Broad usage of the term “dialectic” includes meanings of both dialogue and logic. For Plato, dialogue aimed directly at truth (though not necessarily reaching it). Aristotle considered a many-sided logical/semantic analysis to be the single most important tool of science, and more rigorous than the dialogue that was Plato’s favorite literary device.
For Aristotle, unlike Plato, dialectic is not a direct quest for truth. It is instead an inferential/semantic examination of opinion or what is merely said (or, I would argue, of appearance). It uses the same logical forms as the rational knowledge Aristotle called episteme; but unlike the latter, yields results that Aristotle calls only “probable”, because they depend on premises that are merely “said” rather than rationally known. (This is a qualitative assessment having nothing to do with statistical probability.) This has often been taken as a denigration of dialectic. I take it instead as an affirmation of the importance of semantics. (Plato would already emphasize that it is a matter of an ethically motivated quest for truth rather than a claim to mastery or simple possession of it. Aristotle opens things up further by preferring an indirect, semantically oriented approach to the quest.)
Aristotle also says (Topics Book 1) that dialectic in just this “merely probable” sense is the best means we have for getting clarity about first principles. Aristotle’s own approach to what later came to be called “metaphysics” is (“merely”) dialectical in a specifically Aristotelian sense. In being so, it is essentially semantic and normative. I don’t think Aristotle regarded metaphysics as episteme (“science”) any more than he regarded ethics or phronesis (“practical judgment”) as episteme, and in neither case is it a denigration. (Aristotle is far more honest than most later writers about the relatively less certain nature of so-called first principles, compared with many other apparently more derivative results. He is the original antifoundationalist.)
Hegel actually says the greatest example of ancient dialectic was Proclus’ commentary on Plato’s Parmenides. (He did not know the work of the other great late Neoplatonist Damascius, which was even more sophisticated). Parmenides explicitly examines a series of antithetical propositions, which does resemble the common image of Hegelian dialectic.
While I think one should absolutely not try to read the common image of Hegelian dialectic into Plato and Aristotle, very fruitful results can be obtained going the other direction. I particularly see a lot of Aristotelian-style dialectic in Hegel. Underlying the occasional emphasis on antitheses is a broader concern for actually many-sided inferential/semantic examination of opinion or appearance.
My own candidate for the greatest example of ancient dialectic is the development of the concepts of ousia (“what it was to have been” a thing) and energeia (“at-work-ness”) in the central books of Aristotle’s Metaphysics. As in the biological works, merely binary distinction is not the main point there.
The stereotype of a binary schematism at work in Hegel is not without basis, but more careful commentary has limited its scope. Aristotelian dialectic actually pervades Hegel’s works.
In a dialectical development (Aristotelian or Hegelian), it is common to begin with one presumed meaning for a term, and end up with a different one. The classic discussion in the Metaphysics mentioned above begins with the idea of a simple substrate that remains constant through a change, and goes through multiple transformations to progressively richer concepts.