If you write Democrat or for that matter Republican with a capital letter, that refers to political parties. If you write democrat without a capital letter (as MAV seems to prefer), it just means anyone who believes in democracy, while republican means anyone who is in favour of not having a hereditary monarchy.

Snce both those parties are, to the best of my knowledge, in favour of democracy and opposed to hereditary monarchy in the USA, using the lower case doesn't really seem helpful.