I think there's a bit to unpack in this question, starting with: Were the Americans actually successful against the British navy in the war of 1812; and if so, were they more successful against the British in 1812 than they were against the French in the quasi-war of 1798-1800?

To the first question, I would argue that the American navy was not in fact particularly successful against the British navy.

If we take as granted that the functions of a navy are to control the seas and rivers of its nation, to project power across the seas, and to protect shipping and commercial interests of the nation, the US Navy did very little of that during the war of 1812.

The single-ship actions fought by USS Constitution -- the capture of HMS Guerriere, HMS Java and HMS Cyane and Levant were significant as single-ship actions, but negligible in the larger scope of the war. (It's worth noting that Constitution captured Guerriere only after it was forced to be towed using its own ship's boats to escape an overwhelming British squadron.) You also have some other single-ship actions, such as USS United States capturing HMS Macedonian, or USS Wasp capturing HMS Frolic (then immediately being captured itself, but foundering). Set aside those, though, are also British victories over United States ships -- Shannon vs. Chesapeake,, Phoebe vs. Essex, and the like. And there are many examples of "asymmetrical" captures, that is larger ships capturing merchant ships or smaller ships during the war. (Stephen Decatur lost USS President to a four-ship British squadron after running her aground outside of New York harbor.)

There was a real effect to those battles, to be sure -- the single-ship actions of the small American navy inspired morale at home -- but an American fleet based around six large frigates couldn't do much set against a British navy of 600 ships. And in fact we saw that during the course of the war. The American navy was unable to prevent its ports being blockaded (though the British permitted northeastern ports to trade grain to Canada, in part to fuel sectional discontent). The British also famously raided throughout the Chesapeake, extorting protection money from American towns and burning several of them, most famously Washington, D.C. Numbers of merchant ships captured are hard to come by, partly because many were captured and then recaptured nearly immediately, but the Americans may have lost up up to 1,400 while Lloyd's reckoned British losses at 1100 or so -- this crippled American trade, but affected less than 10% of the British merchant fleet.

So ... no, by any sensible measure, the Americans were not successful in the war of 1812. Though the American frigates also had some success in the Quasi-War, a major difference between it and the war of 1812 is that half of the six-ship Navy didn't exist in the Quasi-War; Chesapeake, Congress and President weren't launched until late in 1799 or 1800, and cruised mostly against merchants.

Well lets break this down..

Were they? And in what circumstances were they?

I would frankly point out that by and large the USN wasn't particularly more effective outside of a few lucky single ship engagements, bolstered by their intentional choices towards force structure and ship building. And by 1814 were about as beaten down as the French Navy by the time of Napoleon's abdication by and large.

We can even just look at the fates of the Original 6 Frigates, by the end of the War of 1812 a pair had been captured, President and Chesapeake. Another never made a cruise during the war, Constellation spent the war slowly being repaired at the Washington Navy Yard then stayed at anchor to bolster the defenses at Norfolk to protect the Navy Yard there. United States with a repaired Macedonian had tried to break out of New York but were forced to seek shelter at New London where they would sit for the last 18 months of the war, their crews were even sent north to the Great Lakes to aid efforts there. While Congress had a thoroughly mediocre series of cruises and spent the last 18 months of the war sitting in ordinary at Boston as well. When the war ended the only one of the sisters at sea was the Constitution off Brazil. The cost of provisioning a frigate for a cruise, repair of battle damage, and the tightening British Blockade simply made risking the ships a losing gambit in most circumstances. As Decatur found out the hard way when he tried to get the President to sea out of New York harbor and was captured for his trouble.

But that is not to say that the USN, with little real prestige among the eyes of the naval establishment of the time didn't shock the world with some of their victories. In particular the 3 frigate actions in the second half of 1812 absolutely did put the Admiralty in a tizzy to respond. The Constitution had defeated 2 British ships in different actions and the United States one as well. But we can absolutely say these were far from "fair" fights as it were, as the US ships were 2 of the 3 "Super-Frigates" then in service. Designed to fill the hole left by the nation's unwillingness to attempt to build and operate ships of the line, these giant frigates routinely carried 50 or more guns. And were far larger in size than their counterparts.

The French meanwhile had no real comparable class of ship in their wars with the British. And in general also tended to do things like build ships using green timber and focus on getting ships to sea as quickly as possible. The US having a relatively small force also has the ability to man ships at least for the officer corps with only those on the lists who had shown real ability for the important commands(the frigates), Stephen Decatur's rise is a perfect example here. While the Revolution and years of attrition had done much to shrink the bench of competent naval leadership for the French compared to their foes.

But in the end the British adapted to the threats posed by both. Enforcing stricter orders on their frigate captains in American waters, and even razing a few older ships of the line to serve as a direct counter to the USN 44's. Both had success in single ship actions, and managed to get raiders to sea that played havoc on shipping, but both also had their ports blockaded and were suffering a slow strangulation of their shore establishment and numbers.