"Demotic" simply means "common" or "popular", which is why it is sometimes used to describe the simplified colloquial Greek used in everyday life (though it should really be spelled Thimotiki with a hard "th" as in "this" to pronounce it correctly), which distinguishes it from Katharevousa, the artificial language that has been used in official Greek documents (but which is still really not all that elegant, when compared with proper Classical Greek).

Demotic, in the world of antiquity, is a dual-use term that describes (1) the late stage of the Egyptian language that immediately predates Coptic, which was used, roughly, in the Ptolemaic period (actually, from c. 500 BC to c. 200 AD) and (2) the simplified cursive writing system used to record texts written in that form of Egyptian.

I've never seen it capitalized or not to distinguish it from common modern Greek, but then again that's probably because people reading books on Egypt aren't going to immediately ask, "Oh wait, did they mean modern Greek or late Egyptian?"