I always thought of the Sharmat as Dagoth, and the Hortator as Nerevar. But there's more to it, right? Can someone fully explain the concepts of Sharmat and Hortator?

A Hortator is a war leader elected by all the houses (or maybe just the relevant ones given that you only had to convince three houses in Morrowind). Presumably this prevents fighting over who gets to lead all the armies.
Sharmat isn't really explained, it probably means something like devil, false god or opponent. Dagoth was not the Sharmat till he became a god.

The fun part about the Sharmat comes from reading http://www.imperial-library.info/content/morrowind-thirty-six-lessons-vivec-sermon-fifteen. Parts of the poem in there make the Sharmat sound more like Lorkhan, or at least his heart, than Dagoth Ur, though the latter is certainly also referenced; perhaps it's hinting at some deeper connection between the two due to Dagoth Ur's more intensive use of the heart's powers. That's my interpretation of it, at least; whether that is accurate or meaningful, and how much, is, of course, strongly up for debate or disagreement. <_<

The Sharmat is the mirror, is the reverse of the coin. Both are trying to defend their people and drive out an invading force. The only difference between the two is that the Sharmat is not the Hortator.

That's how I understood it.