The Brits did not respect US independence. A war was inevitable. The final decisive US victory at New Orleans earned the US respect and British acceptance of the Louisiana Purchase. If the British had captured New Orleans, the outcome could have been very different.
How do you reckon that? It is well known that the treaty of Ghent was signed before the Battle of New Orleans, and you are saying they would have said ‘Oh, on second thoughts....’
In any case, a few days after the Battle of New Orleans the same British force (under more effective new leadership) successfully captured Fort Bowyer and were within an ace of capturing Mobile, Alabama when news of the treaty arrived, whereupon they immediately vacated the field and the fort, the same would have happened had they received the message had they successfully captured New Orleans.
IMHO the spectacularly one-sided nature of the Battle of New Orleans has been seized and contorted by biased historians as being more significant than it actually was, and is used as a distraction from the fact that the US did not achieve it initial aim of conquering Canada and ended up having to spend the rest of the war on the defence trying to stop the British rampaging through the US...
The Battle of New Orleans was fought after the war was over; news of the Treaty of Ghent had not arrived. Even if the British had captured New Orleans the would have withdrawn.