The War of 1812 was significant because the Brits accepted US independence and the Louisiana Purchase, both of which would have been in jeopardy had they won the Battle of New Orleans.
Yes, the British considered their victory over Napoleon to be decisive. They put the old Royals back in power in Europe. However, that did not last very long.
First off, the Battle of New Orleans was fought after the Treaty of Ghent had been signed and changed nothing about the terms. As for U.S. independence, that had been accepted by the British in 1783, so there was no change from before the war, and I can't recall ever seeing the British contest the Louisiana Purchase. Certainly the Treaty of Ghent makes no mention of it.