This question is a tricky one to answer because you seem to be assuming a few different things that don't quite hold up.

First of all, the idea that the United States defeated Great Britain in the War of 1812 is pretty tenuous. They did successfully fight them to a draw in that Great Britain wasn't able to occupy large parts of the United States, and at the negotiating table Americans were strikingly successful (it helps that the British had sent their best negotiators to deal with events in Europe re: Napoleon; the Americans didn't have to rely on their B Team as it were). Never again were the British quite so willing to contemplate war with the United States. But the invasion of Canada had gone rather badly for the United States, exposing weaknesses in the armed forces. Northern Massachusetts (now Maine) was occupied, and Washington was burnt. Americans did a good job of distracting themselves from the blockade with individual naval victories, none of which materially changed the fact that the blockade slowly starved America's merchants. Furthermore, the national economy was rapidly being wrecked by the war; the U.S. government faced near-bankruptcy by the end of the war, and the suspension of payments in specie (gold) meant that currency in the United States risked soon becoming dramatically devalued. All of this happened as the war exposes ugly regional divisions in the United States. While much of the earlier discussions among historians that New England was on the verge of seceding away from the United States appears to have been overblown, New England was nevertheless profoundly discontented because of the war, and trading with the enemy in order to stay alive.

As I mentioned, to the extent that the United States wins the war, it's at the negotiations in Belgium. Great Britain gives up the territory it had captured in Michigan and Maine and more or less abandons its direct support for the Native American tribes it had allied with during the war. Those tribes had fought for the British, and without support from their ally are rapidly defeated in the course of a few years, opening the west to further expansion.

In 1815, the United States had a fundamentally different economy and status than Great Britain. It had won a draw against a divided Great Britain and was still hit hard by the war, and Great Britain was seeing the takeoff of the industrial revolution as it became an economic superpower. The United States simply wasn't at that point, and industrialization would take several more decades before the United States would begin to surpass the British.

More broadly, few if any Americans really wanted to take on that kind of great power status. They felt they had at least a degree of security from Great Britain, so there was no pressing need to build up militarily. Donald Hickey makes much of the fact that Jefferson had tried to avoid military buildups before the war, and in the wake of the war, a kind of balance is struck, with a navy large enough for defensive purposes, but not enough to challenge the British dominance of the seas. With the defeat of Napoleon and the end of impressment, there was little needed for anything bigger than that. They faced no other serious threats, and with the defeat of Napoleon, the international order would be fairly stable (for Europeans) for decades to come.

More to the point, the United States had removed the last serious barriers to westward expansion. The tribes opposing them were defeated in due course, and while the Spanish (and in the next few years, Mexican) government held land in the U.S., they were relatively weak and facing a much stronger United States.

Lastly, to your point about Spain, it wasn't the defeat of Spain inherently that revealed the United States as a world power, so much as its outward-looking orientation. Among the great powers, Spain was functionally irrelevant, and most observers were at least aware of the fact that the United States had a large, sophisticated economy that was outproducing its competitors by a substantial margin. What changes is how Americans look at the world. For almost the entirety of the 19th century, the United States had focused on continental expansion, and its two largest wars after 1812 were one with Mexico, and a civil war. In 1898, suddenly the United States is looking at Cuba, and on top of that, it takes the Philippines, a region which hadn't even been part of the original dispute between the United States and Spain. It suggested a new orientation on the part of American leaders (one which was, itself, not consistent; after the horrors of the Philippines War, many Americans doubted whether it had been a good idea).

Like, the United States defeated Great Britain in the War of 1812.

You are starting from an erroneous base assumption. The United States did not win the war of 1812.

From the point of view of the Britain, the US lost the war of 1812. It can only be considered a success for the US in that one of their war goals was prevention of British impressment of American seaman onto their naval vessels. That was not its only war goal though. It quickly became a reason to gain territory by taking Britain's other north American colonies.

From the point of view of Britain, it successfully defended its colonies primarily with colonial militia units, even though the bulk of its military forces were focused on the Napoleonic wars in Europe. The war came to a pretty quick end once the Napoleon situation was dealt with and they could start sending more units of veteran regulars over to deal with the United States.

In the end, the war is often thought of as the war that no one won. No territory changed hands, no reparations were paid, and everything returned to status quo antebellum.

Considering that Britain was able to hold the United States off in an essential stalemate with mostly local colonial militia units, it is kind of clear why the United States would not have been considered as a significant world power as a result.