historically? constitutional republicans wanted to replace monarchies with laws rooted in a special document. they were fundamentally legalists and never wanted the French revolution to carry on past the abolition of the monarchy and cheered Bonaparte's counterrevolution. democrats wanted the vote - the term eventually came to refer to those who wanted universal(*) suffrage. universal here doesn't mean actually everyone. only their most radical wing wanted suffrage for women or immigrants. rather, universal referred solely to class - instead of restricting suffrage to landholders, they wanted to extend it to all male citizens, regardless of class. their most radical elements wanted genuinely universal suffrage and more - they weren't content with the political revolution of the French revolution and wanted a social revolution to accompany it and to bring true freedom to all. these were the first socialists, or social democrats as they styled themselves then, in the century following the revolution.

that is, the American political parties named themselves after the liberal wings of the French Revolution. these terms have been meaningless since the first world war, when the bourgeoisie completed the transfer of power from the old aristocracy to themselves. or as Marx put it:

The tradition of all dead generations weighs like a nightmare on the brains of the living. And just as they seem to be occupied with revolutionizing themselves and things, creating something that did not exist before, precisely in such epochs of revolutionary crisis they anxiously conjure up the spirits of the past to their service, borrowing from them names, battle slogans, and costumes in order to present this new scene in world history in time-honored disguise and borrowed language. Thus Luther put on the mask of the Apostle Paul, the Revolution of 1789-1814 draped itself alternately in the guise of the Roman Republic and the Roman Empire, and the Revolution of 1848 knew nothing better to do than to parody, now 1789, now the revolutionary tradition of 1793-95.

they're trying to revivify the dead, to wear the costumes of epochs past in the hopes of conjuring up the spirit that had animated those revolutions and unified the people. where the revolutionary bourgeoisie had worn the guise of Rome, the reactionary bourgeoisie of the present try to conjure up the French and American counterrevolutions, trying to sweep the footnotes that became the centerpieces of history in the twentieth century back under the rug in the process.