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This is a good question, and as you can imagine, the answer is quite complicated. But we can unpack it a bit, and speculate on some of the larger, thornier results of the war that arguably had a much larger impact on the trajectory of North American history than usually gets examined.

So first, we'll take a look at the war aims of both sides, and whether those aims were achieved at the end.

Then we can examine the aspects of American and European policy that directly rested on the end of the war, even if their effects were not immediate.

I will then spend a bit of time talking about the state of historiography of the war, and the problematic use of personal memoirs as a touchstone for modern historians, and how the war is popularly interpreted in a nationalistic sense, chiefly by those in the US and Canada.

1. War Aims

First, the causes, as articulated by Americans.

The United States declared war on Great Britain on June 17th, 1812. President James Madison's address to congress to call for the vote to declare war stressed three major causes that, Madison claimed, made war the last resort of injured nations.

1. Impressment. British cruisers have been in the continued practice of violating the American flag on the great high way of nations, and of seizing and carrying off persons sailing under it. He was describing the British practice of Impressment: the deliberate searching of neutral nations' vessels and plucking off British citizens and especially former Royal Navy sailors who had deserted. The US felt particularly targeted, and the seizure of American sailors to fight the British war was considered a heinous insult to American sovereignty. I go over more detail on one particularly aggravating episode in this post

2. Trade rights. Pursuing their war against Napoleon, the British had made a habit of plundering American trade. Their "Orders in Council" forbade all trade to the European mainland apart from those trading at British ports. Madison expressed a popular sentiment when he claimed that our commerce has been plundered in every sea. The practice had, according to those who supported the war, wrecked the US economy.

3. The "Indian Problem." ...Our attention is necessarily drawn to the warfare just renewed by the Savages, on one of our extensive frontiers; a warfare which is known to spare neither age nor sex, and to be distinguished by features peculiarly shocking to humanity. The US perceived a Native-led spiritual revitalization led by a former wastrel who had changed his name from Lalawethika or "The Loudmouth/Rattle" to Tenskwatawa or "The Open-Door," who was often referred to simply as "The Prophet," as a British-led conspiracy to ignite the frontier against the United States. Andrew Jackson voiced a popular theory that The Prophet as his charismatic brother Tecumseh as an engine set in motion by the British.

These were the three primary causes of the war from the US perspective, and these are in no way comprehensive, and does not express the myriad caveats and complications with these claims: they are simply the popularly held opinions of those pro-war. It's outside the scope of the question to delve into the many problems with these opinions.

The British war aims were much clearer: Protect the territory of Great Britain, her citizens and dependents, and continue fighting the war against Napoleon.

The war aims, for the United States, was to end impressment, end trade policies that brought economic ruin to the United States, and end the "Savage" war on the frontier. The vector for accomplishing this was a vigorous naval war against British shipping and nothing short of the conquest of Canada. Every pro-war voice agreed that Canada must be conquered, but there were highly variable ideas about what to do with it afterward. Some wanted to annex it into the United States, citing the failure in the War for Independence to bring Canadian colonies into the war. Some wanted merely to exchange conquered territory back to Britain in return for favorable trade policies. Still others assumed that the Canadians would be so happy to be "freed" from the "shackles" of British tyranny that they would voluntarily attach themselves to the United States. Various historians have stressed particular aspects of these beliefs and have said "X was the plan" but the reality is there was no plan. They'd conquer Canada - everyone was fairly confident about that, and Jefferson himself notoriously quipped that the conquest of Canada was a mere matter of marching - and then figure it out afterward.

The results of the war you likely already know. The US failed repeatedly to conquer Canada, ended up blockaded on the east coast with lone frigates cut off from supplies and ammunition, and on the verge of disunion. To simplify things a bit, a few high profile, skin-of-the-teeth victories in late 1814 gave leverage to end the war on favorable terms. This, too, was a consequence primarily of European affairs: the British diplomats had little autonomy and the US sent what can be regarded as their first-string of diplomats, including Henry Clay and John Quincy Adams. With Napoleon's first defeat came the possibility of renewed and far more sustainable effort on the British side to continue the war, the US was quick to agree to terms they found relatively favorable.

Impressment was a non-issue, with Napoleon (temporarily) defeated. No more need to man the Royal Navy on a war footing meant that impressment was ignored.

The Orders in Council had actually been repealed before the war was declared, and was again a non-issue, since fighting Napoleon was the catalyst of that particular policy anyway.

The "Indian problem" was the only leg of the war in which the US had achieved unambiguous success. I don't want to suggest that it was in any way easy or even a forgone conclusion, but the death of Tecumseh and other influential firebrand Native leaders allowed the new-born confederacy to dwindle in power and influence, and the decentralized political structure of native groups and bands started a long process of peace-making with the United States.

2. Effects

One consequence of the successful end of the Indian war meant that, though the British never really articulated a particular policy as a result, they backed off from North American almost entirely. What this meant was that the Native groups could no longer rely on relatively balanced political and military power between the US and Great Britain, and the sole monopoly of political power in the Great Lakes region especially was the United States. It took a decade or so for this to become clear, but is unambiguously the major influence on North American history in the first decades of the 19th century. It is the basis for nearly every territorial decision with respect to Native territory and autonomy going forward.

Because of the popular sentiments at the end of the war about the battles of Plattsburgh, which ended a potential British thrust into the political heart of the United States, and New Orleans, which accomplished nothing but raise US morale after the war ended also stifled the many, many voices that had argued against the war in the first place. The media spin about these battles essentially represented the war as a major victory, and popular fervor suppressed the counter-arguments to the Jeffersonian political approach.

This is a much harder result to trace, but in effect, the end of the war led to the ruin of the Federalist party, and its successor political party was in constitution and political goals a totally different party with a much different makeup. The Washington/Knox/Adams leg of the party, and even the Hamiltonian financial structure, was swept away into a much more economically-facing party that eventually represented the interest of the eastern elites in a way that the old party never really did. This also dampened, initially, any support the Federalists had ever had for abolition. Later iterations of the party of the opposition (there were many of these parties and their only real commonality was resistance to the majority-in-power, the framework and context of the parties were all different) changed, of course, and led, eventually, to the Republican party, and we all know what that meant.

I don't want to lean too hard on this point, because a lot of this is something that I have seldom seen expressed in a comprehensive way, but essentially the war was a political bottleneck for the United States, and the perception of the war as wildly successful strained out many of the voices in the United States and elsewhere that were advocating an alternative of the Jeffersonian political model. This led to a renewed aggression in land acquisition and all of the nightmarish elements of that industry, consolidated support for slavery and set back abolitionist voices by at least a decade, and fractured the opposing voices into smaller, less cohesive wholes. This is not a good or bad result, merely a kind of cultural colander that only allowed a particular expression of power to remain while disrupting those that had disagreed since the founding of the nation.

This is all to say that, though the United States failed to achieve nearly any of its stated goals, it did lead to the US and the 19th century as we know it. Without going into counter-factual territory, small changes to the results of certain votes, battles, or whole campaigns could have potentially radically altered the history of not only the United States, but North America as a whole - and subsequently the history of the entire hemisphere.

From a big-picture strategic viewpoint, it was a draw. Neither of the major powers really got what they wanted - the US tried to seize Canada as a bargaining chip to negotiate better terms with Britain, the UK tried to create an "Indian Barrier State" between the US & Canada to safeguard their territory. But neither really lost anything except for manpower. Both sides came away with victories they could be proud of, and the Treaty of Ghent restored everything to a status quo ante bellum.

The only clear losers were the Native American tribes, who largely sided with the UK and were defeated handedly by the US, particularly in Andrew Jackson's Creek Campaign. After the war, the US claimed tens of millions of acres of new land from the Native Americans, even from tribes who had allied with them.

However, it does seem to be the prevailing opinion in the US, UK, and Canada that they each won the war, and for understandable reasons. As you mentioned, Canada successfully and repeatedly repelled US invasion attempts, which had been the major war aim of the US. The UK shares that success and also has the blockade and the burning of the White House to triumph.

From the US perspective, they repelled British invasions of New England and Louisiana, had some iconic (if minor) naval victories, and gained lots of new land from the Native Americans. But most notably, the Battle of New Orleans left a lasting impression in the US psyche. It was the last, largest battle of the war, and was a decisive US victory over a superior British force that was attacking US soil. It ended the war on a high note, so to speak, and helped lead to a decade of renewed national unity.

As for whether the war was necessary or not, it really comes down to perspectives on the impressment of US sailors, which probably deserves a question of its own. But suffice it to say that the US government at the time felt that its neutrality was being violated and it had to do something to respond. After diplomacy and economic sanctions failed spectacularly, the Madison administration decided to declare war in an effort to gain a bargaining chip to force the British to come to the table. They were unable to do so, and when the British eventually stopped the practice of impressment on their own in 1814, after Napoleon was defeated, it made continuing the war unattractive to both sides.