The war of 1812 was basically a draw. Further, it was a pretty big blow to American confidence (bear in mind, for instance, the British burned the White House when they took DC).

But after the war, the US economy really started booming after the war ended. The US is - in a short period of time - on a reasonable decent footing in terms of military and diplomatic strength.

But yeah, War of 1812 is more or less a wash out for the British and the United States. If there were losers, in a sense, it was the approximately 10,000 Native American allies for the British, who lost a lost of leverage with the US gov't by siding with the UK in the conflict.

Can you clarify what you mean by "culturally gaining" something?