Much of the answer to this lies in the fact that Enlightenment ideals, which had been a stimulus to the American independence movement, were also a source of controversy and division in the newly formed country.
The Enlightenment had promoted not only the ideal of human equality, but a break with tradition as manifested in different areas of human thought and social organization. The main question as the new constitutional government was initiated in 1789 was, how far did Americans wish to go in their creation of a new society? Had a merely political, not ideological separation from Britain occurred? Or did the events of 1775–83, and then 1787–88 during the constitutional settlement, mean that a true overthrow of past dominant ideas had taken place?
The debate over these issues was intensified largely because of what was occurring in Europe. The French Revolution carried with it implications about the future not just of France, but of the world. Did it, as many believed, represent a fulfillment of the same ideals that animated the American Revolution, which France (though still an absolute monarchy at the time) had facilitated? Or was it a dangerous, radical step that was aimed at subverting traditional values and was thus a danger to the order of civilization?
These questions and the division in thought they represented were at the roots of the nascent US political parties in the 1790s. The Federalists, with their chief ideologue Hamilton, who was Washington's right-hand man, wished to maintain a strong central government and an economy grounded in a central banking system. They also intended to restore relations with Great Britain. This orientation became even more pronounced when the French Revolution turned increasingly violent and radical, not just forming a constitutional monarchy but eventually deposing and executing Louis XVI.
The Democratic Republicans under Jefferson's leadership wanted a weaker central government and an economy based chiefly on yeoman farming. They supported revolutionary France and regarded the monarchies of Europe, including that of Britain, as reactionary states that would eventually be replaced by republics, as France had become in 1792. In all, though this is an oversimplification, the Federalists were "conservative" and the Democratic Republicans were "liberal," though these terms were not used in that time as they are now.
There was also a religious dimension to the American ideological split. Jefferson was a deist: like the Jacobins in France, he rejected organized religion. Though this was probably true of only some of Jefferson's followers, it was a tendency that existed as a kind of subtext to their general beliefs. Jefferson's friend Thomas Paine wrote two books during the period that were expressive, respectively, of social and religious (or anti-religious) radicalism: Rights of Man (1791) and The Age of Reason (1795). These works were highly inflammatory and caused conservative Americans to regard Paine and anyone who favored him (such as Jefferson) as a hateful symbol of the rejection of cherished tradition and religious belief—two things most Federalists did believe in. By most standards, the "culture war" of the 1790s was worse than the similar disputes between conservative and liberal ideologies in our own time, but it did set the course for the kind of political debate still with us today.