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I think I've found a paralelism between these two notions, at least to some extent. For Kant defines (in Logic, I, I, §1, also translated) concept as "an universal representation"

Every knowledge, i.e. every representation referred with consciousness toward an object, are either intuititions [Anschauungen] or concepts [Begriffe]. The intuition is a particular representation (repraesentatio singularis); the concept, a universal representation (reprepraesentatio per notas communes) or reflected representation (repraesentatio discursiva). [my poor translation]

While Hegel defines essence (Wesen):

The terms in Essence are always mere pairs of correlatives, and yet not absolutely reflected in themselves: hence in essence the actual unity of the notion is not yet realised, but only postulated by reflection. Essence – which is Being coming into mediation with itself through the negativity of itself – is self-relatedness, only in so far as it is relation to an Other – this Other, however, coming to view at first not as something which is, but as postulated and hypothesised.

Being has not vanished: but, firstly, Essence, as simple self-relation, is Being, and secondly as regards its one-sided characteristic of immediacy, Being is deposed to a mere negative, to a seeming or reflected light – Essence accordingly is Being thus reflecting light into itself. [Enz. Logik, §112, quoted as this translation]

The addendum sometimes is a little bit more friendly than Hegel; here:

Wesen in ordinary life frequently means only a collection or aggregate: Zeitungswesen (the Press), Postwesen (the Post Office), Steuerwesen (the Revenue). All that these terms mean is that the things in question are not to be taken single, in their immediacy, but as a complex, and then, perhaps, in addition, in their various bearings. This usage of the term is not very different in its implications from our own. [from here]

What I think about this is that the Kantian definition of concept coincides here with the hegelian definition of concept (or Notion in that translation); meanwhile Hegel sees the essence as "the concept in relation to itself", Kant calls Begriff the "reflected representation".

Off course, this similitude must not be taken for granted, since every step in Hegel's logic go far beyond what normally we call "logic"; if I'm right, this similitude is restrained to a very specific sense, to the meaning of the essence for the understanding (Verstand).

What do you think? Am I overinterpreting this?

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+1 for a really interesting, but difficult question to answer. Hegel's notion of a concept differs in that "every concept [...] gives rise to antinomies or contradictions" (except the Absolute Idea) according to Wikipedia's article on the Science of Logic which might be a good starting point, en.wikipedia.org/wiki/Science_of_Logic –  Aputsiaq Mar 16 '12 at 4:38
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They're different but related. From the SEP entry on Hegel

Like Kant, Hegel thinks that one's capacity to be “conscious” of some external object as something distinct from oneself requires the reflexivity of “self-consciousness,” that is, it requires one's awareness of oneself as a subject for whom something distinct, the object, is presented as known.

This 'awareness of oneself as a subject' is the what Hegel mentions as 'Essence – which is Being coming into mediation with itself through the negativity of itself – is self-relatedness, only in so far as it is relation to an Other'

But Concepts for Kant is simply that which makes Intuitions intelligible to the Understanding. This involves the above idea as the quotation from the SEP notes, but is separate from it.

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