What are the philosophical (ideological) differences over specific issues during the 1789–1799 period that lead to the formation of political parties?

It's difficult to answer this question without placing the newly formed United States in its international context. The first decade in which the US existed under the Constitution was also one of huge turmoil and upheaval in Europe.
The French Revolution, starting in 1789, had political, religious, and social outcomes with consequences for the entire world. The US was divided in its reaction to what occurred in Europe. Washington, Hamilton, Jay, and others who formed the nucleus of the Federalists wished to regard our War of Independence as having been a family quarrel of sorts with Britain and to establish a treaty with what had, after all, been the Mother Country. Though France had been our first ally, Washington grew increasingly alarmed at the disorders and violence occurring in Paris which culminated in the Reign of Terror. Thomas Jefferson, on the hand, believed in the French Revolution as a kind of absolute, in which the ends justified the means by which a complete social transformation would be carried out.
There were economic corollaries to these different positions among the American leaders. One side wished to have a stronger central government in which credit and commerce were controlled by financial institutions, mostly in the Northern US States. This was the position spearheaded by Alexander Hamilton, who was essentially Washington's right-hand man. In this view, the separation from Britain in 1775–83 had not been a social revolution, and the traditional principles by which the monarchies of Europe, including Britain of course, were still valid. By contrast, Jefferson and what became the Democratic Republican party seemingly wanted to create a different type of society based chiefly on yeoman farming and the (relative) lack of centralized control in the US. Jefferson saw this as being in accord with the Revolution enacted in Europe, in which the French revolutionaries sought not only to create a new kind of society in France, but to spread their ideology of social change to the other European countries, overthrowing the monarchies and turning them into republics.
Add to this the issue of slavery. Hamilton and the nascent Federalists were clearly anti-slavery. Washington, though he continued to hold enslaved people, freed all of them in his will, something none of the other leading Founders from the South had done. Jefferson, though he spoke and wrote against slavery, freed only a handful of enslaved people (all of whom were related to him personally) and did nothing to encourage the eventual abolition of the institution. His view of social revolution did not include the African Americans, of whom he envisioned a kind of benign deportation to Africa or the West Indies. Though slavery during the period 1789–99 had not yet become a central issue for US elites, the party system of Federalists (who were eventually re-organized as the Whigs and then as the new Republicans in the 1850s) vs. Democratic Republicans (who became the Democrats) was the basis of the split in the US in the nineteenth century, which led to the Civil War.