The biggest change that occurred in this time was the French Revolution. France became the ally of the US in 1778 (shortly after the battle of Saratoga) and the quasi-war took place during the Adams administration (1797-1801) while the French Revolution began in 1789, and the French monarchy fell in 1793. The French allied with the US while under the rule of King Louis XVI, mostly in an attempt at vengeance towards Great Britain (an absolute monarchy normally would have no incentive to support a republic that would threaten monarchies worldwide). France, specifically Louis XVI's uncle Louis XV, lost the Seven Years War to Britain and was looking for rectification. However, the alliance resulted in millions of French francs being spent on the war, bankrupting France. This war, coupled with existing debt from the Seven Years war and other reckless expenditures of the monarchy was a large reason for revolution and, naturally, once the revolution succeeded the new government had a bit of animosity toward the US. In addition to this, the Federalist government of Adams preferred Britain and signed treaties with them (Jay Treaty), but not France. This gave favored trade relations to Britain and angered the French. This all bubbled over when the US refused to pay debts it owed to France from the war, leading to the XYZ affair, when three French diplomats stated to the US government that they would restore relations with the US provided they pay a significant bribe. This pretty much turned the public's opinion of France hostile, thus spawning the phrase at the time "millions for defense, not one cent for tribute!" (suggesting that the people would rather spend millions on ships and armies than pay a cent to the French diplomats).
http://en.wikipedia.org/wiki/Jay_Treaty http://en.wikipedia.org/wiki/Causes_of_the_French_Revolution#Economics_and_finances
EDIT: added more detail.
While the XYZ affair and the refusal to pay debts caused some international tension, I really don't think the French were ever a rival to the US. I mean, that was scarcely a bump in the road.
France is America's oldest ally, they fought together in World War's one and two, America intervened in Vietnam to aid France and the two nations have been economic allies for over 200 years.
http://en.wikipedia.org/wiki/France%E2%80%93Vietnam_relations