The classic whale vs. the wolf scenario.
The British lack the manpower and infrastructure to threaten the USA in North America. While they have a great many resources they could throw into a war, they face the same issue every global superpower does, that they spread those manpower and resources globally with a number of threats and rivals keeping them from tossing too many of them into one theater.
Long story short, the British could on paper make this war very painful for the USA, but only if they're willing to make it very painful for themselves elsewhere. London is not going to trade Cape Town or Bombay for Seattle or Boston for the take of Toronto.
Thus, they are left to dealing the USA some painful body blows, but are held back from dishing out anything close to a knockout.
The Americans have a colossal advantage in North America in terms of everything from manpower to railroads, and are more than capable of overrunning damned near the whole of Canada without more than token resistance.
Sorry Canadian posters who always say that Canada somehow beats back the Americans like a hockey-playing Vietnam. A country of 5 million people, most of which is barely settled farmland, is not fending of an industrial militarized nation of 80 million that is on the other end of the border.
Outside of North America though, the USA likely kind bring too much against Britain, especially the farther you get from North America. US Marines flying the Stars and Stripes over Parliament is as absurd as the Redcoats taking New York.
Thus, while the USA is more than capable of swatting aside anything standing in their way in North America, they lack the ability or capability to do much to the British Empire at large.
Here's the kicker though - both sides of the conflict, Americans and British alike, have heavy commercial interests on both sides of the conflict. No matter how loudly the warhawks may scream in DC and London, the tycoons and the merchants will scream just as loudly to end it quickly.
Thus, rather than a stunning victory for either side, the war likely ends in the two coming to an impasse.
The USA likely is ceded some less important chunks of Canada, likely the underpopulated Western and Northern bits, in exchange for damages and some trade concessions being given to Britain. Both sides get to claim victory at very little cost to themselves, and the only real losers are Canadian nationalists.
The impact on North America might be interesting too.
The USA has more frontier on the home front to settle and carve states from - I see the future American prairie states taking a much different shape than the OTL Canadian provinces. Smaller certainly, the size of the Dakotas rather than Texas. Alaska will be connected to the lower 48, which will have an impact on regional settlement, especially with the Yukon and Alaskan gold rushes ready to start soon. The Wild West may be closed, but the Great White North beckons.
What's left of Canada is likely going to be hewed to Britain's hip - my money is on the British divided what's left into two or three dominions, which could take any number of eventual shapes. In a few decades, the nations that come from them could be very interesting as they take shape - hell, maybe they'll end up directly integrated into Britain rather than independant?
From there, I imagine things go mostly ala OTL. The Anglo-American war of 1896 being remembered largely as "a lover's quarrel" in the Special Relationship, meriting a few paragraphs in American in British textbooks explaining how North America got its eventual borders.
Meanwhile, people on this website will wonder what might have been had a single transcontinental Canada survived had the war been averted, sparking furious debate from the site's Gaelic speaking Acadian members, Quebeci nationalists and Americans from Vancouver and Edmonton over the idea's merits or absurdities.
