Demotic was significant in the history of language understanding, as it was one of the three scripts on the Rosetta Stone (along with Greek script and Egyptian hieroglyphics). However, it is really only a font difference from hieratic (or hieroglyphics).
The differences in the glyphs are not small, however, as by the time it is generally dated, more than two thousand years had passed since the origin of hieratic. (For an exaggerated example of how much drawings can drift from the original, see A Sequence of Lines Traced by Five Hundred Individuals.) Demotic is more cursive and can be written more quickly. However, demotic is also more ambiguous: some glyphs in demotic represent two or more glyphs in hieratic.
After Psamtik I reunified Upper and Lower Egypt, Demotic became the official script of the realm. However, Alexander the Great conquered Egypt in 332 and his successors managed to hold on to it until 30 BC, when the Romans took over. Egyptian, and hence Demotic, grew less and less important as Greek and then Latin became more important. However, Demotic held on for a long time: the last writing in Demotic known is a piece of graffiti dated 11 Dec, 452 AD.