Henry Clay, a negotiator at Ghent and one of the greatest political actors of his age, summed up three points that the United States had gained from the treaty of Ghent.
1-The British reputation had suffered at Ghent, not the Americans
2- By standing up singlehandedly to the mightiest power on Earth, the United States had established its national character
3- In the ordeal of war and peace, the republican government born of the Revolution had met the crucial test
In addition I would add a few other points. The powerful Indian nations of the Southwest and Northwest that had been able to play off off rival Western powers against each other for generations lay defeated. The United States was able to acquire large amounts of territory in the Southwest, which can be seen here. Secondly while the United States filibusters had failed in their attempts to acquire East Florida, they managed to hold onto West Florida. The late war victories ushered in a degree of unprecedented nationalism leading to the drastic expansion of Federal power far exceeding the Federalist era of 1788-1800, many of these programs are difficult to explain absent a new emerging American nationalism, which can be read more in The Awakening of American Nationalism: 1815 - 1828 by George Dangerfield. Some of these new institutions and programs included but are not limited to: The Second Bank of the United States ( similar to the first but on more Republican principles), The drastic expansion of the United States Navy in 1817, the first protective tariff in American history to protect growing American industry, Federal support for the colonization society, the first Federal pension. Others have argued that the war increased the pace of liberalization within the United States and increased the pace of the market revolution (a term no longer in favor) see The Republic Reborn: War and the Making of Liberal America, 1790-1820.
To expand a bit, what was the point of the whole affair? Was it really about impressment of American sailors into the British navy, which is the only reason I remember for the war from high school
In short no it wasn't just about impressment. Until 1811 in fact the main point of contention between the United States and Britain had been over trade issues. A cornerstone of Republican ideology for both Madison and Jefferson was that free ships made free goods, and that the free transfer of goods around the world would serve as a carrier for Republicanism. The ongoing wars in Europe had resulted in both France and Britain imposing sanctions on neutral vessels, the United States was one of the largest merchant states in the western world and the largest neutral carrier and the restrictions fell hard on the Republic. Events like the Chesapeake-leopard affair hardened attitudes in the United States towards Britain.Moreover Britain had grown fearful over the dramatic expansion of the American merchant fleet, and while willing to give concessions to smaller carriers like Sweden, was not willing to do so for the United States. You also had British support for the native peoples in the Northwest. Finally political relations between Republicans and Federalists had grown increasingly hostile, and Britain was viewed as popularly supporting the Federalist minority, see America on the Brink:How the Political Struggle Over the War of 1812 Almost Destroyed the Young Republic for the struggle between Federalists and Republicans that is particularly harsh on the Federalist for Anglo-American relations see Bradford Perkins Prologue to War:Prologue to War: England and the United States, 1805-1812.
Did the US "win" the war? I always had the impression that the British mainly couldn't be bothered to fight much (being busy with the whole Napoleon thing) and the victories the US achieved (USS Constitution, Lake Champlain, New Orleans) were not that strategically significant.