The War of 1812 was the response of the fledgling US to British actions that were felt to undermine the sovereignty of the US... impressing naturalized citizens (many who were Irish born), encouraging and supporting native tribes to attack American settlers, a host of actions that were considered attacks on American honor, and- yes- a small faction that encouraged the belief that Canada needed liberating.... also, America has two strong opposing viewpoints - that of New England and that of the Southern states... which culminated in our Civil War.

Madison often quoted what Benjamin Franklin was noted having said following the Battle of Yorktown: "The War of Revolution is won, but the War for Independence is yet to be fought." - wiki
It was viewed as the second war of independence by Americans.

Incursions into Canada most likely hoped to capture and hold hostage territory for future exchange... the Treaty of Ghent resulted in no change in the national boundaries.

The war may have been a draw... but American's felt that they had stood up to a bully and remained standing. A relations with Britain were still strained and unpleasant.

The real breakthrough in bringing respect between our two nations came through the efforts of an American businessman to assist Lady Franklin in her search for the Franklin Expedition. The Admiralty had written off the continued search as futile, but she continued her efforts. The two American search efforts failed to find Franklin... but an American whaling ship did salvage one of the "lost" British search vessel, the Resolute.... which was then purchased and refurbished by Congress to be presented to Queen Victoria on 13 December 1856 as a token of comity.

A gesture of peace and respect did more to improve relations between England and America than one war and several treaties...

And realistically, why would the US fight a war to "conquer" Canada when we had just made the Louisiana Purchase which doubled the size of the country? Especially when the northern states bordering Canada were NOT in favor of annexation.