How did Jefferson and Madison differ on questions of central importance to the American Constitution?

The key debate between Jefferson and Madison (which was generally quite amiable; I've attached a letter Jefferson wrote to Madison which shows very little sign of animosity) was on the question of central authority: How much power should be vested in the central government, versus granted to the people at large? It was in a sense asking: "Just how democratic do we really want this democracy to be?"

Madison was distrustful of pure or direct democracy, and as such sought to establish a system of government that would represent the people, but was sufficiently removed from the direct control of the population as to avoid the possibility of mob rule or "tyranny of the majority".

Jefferson on the other hand believed in democracy all the way, and wanted the system to be as responsive to popular opinion as it could feasibly be made. He recognized that pure direct democracy was not sustainable on a national scale (Yet? With modern communication technology, it might be now!), but nonetheless felt that the closer we could get to that ideal, the better.

While Madison believed that the general population was not to be trusted, too easily swayed by demagoguery and factionalism, Jefferson had faith in the people, believing that if power were extended to everyone it would be used wisely in the common interest of all. For Jefferson the great fear was not demagoguery but corruption and tyranny; he did not worry that the people would make the wrong decisions by accident but that those in power would make the wrong decisions on purpose.

In the end, the Constitution was written as a compromise; while it did grant substantial power to the people---more so than really any system of government that had ever existed on this scale up to that point---it also contained a number of provisions designed to separate decision-makers from the direct will of the people. This is why we have things like the Electoral College, for example; it is also why the Senate was originally elected by state governments rather than popular vote.