Assuming they started at the same point (maybe the split of the continents or the migration of what became the first "Native Americans" to the American continent), why did the Old World end up developing faster? When the first conquistadors set foot in the New World, it's clear that the old world had developed technologically (for example militarily) better than the New World.
According to Jared Diamond's Guns, Germs and Steel, one of the first steps from a hunter-gatherer society towards civilization is agriculture. While agricultural societies appeared all over the world, the old world had a more suitable environment, especially with regards to the grains and large animals that lived there.
The old world had wheat, which is easy to plant, harvest and eat, while the new world had maize (corn), which is not. With regard to large domesticate-able animals, the old world had horses, sheep and cattle, while the new world just had llamas.
The old world could also trade east-west, which meant plants and animals could easily find similar climates (because of similar latitudes) over very long distances, while the new world's trade routes were primarily north-south.
Agricultural production encourages a sedentary society, which in turn leads to population growth, specialization of craft and labor, and a ruling class. Put it all together and you get more and better technology.
Depends on what you mean by advanced. If you mean in terms of metalworking, the lack of easily exploited tin deposits in the Americas means that a bronze age never took off. There was a copper-working culture surrounding the Great Lakes, and it pre-dated the chalcolithic in the old world by a few thousand years, but this lasted only as long as the accessible copper ore did.
On the other hand, the civilizations in the Americas had architecture, science, math and literature to match anything in the Old World. The Incas and their forebears were masters of textiles - they built massive suspension bridges, armor, even boats, from cotton. Along the Amazon and Mississippi cultures used massive earthworks projects and advanced horticultural knowledge to sustain huge cities. So, what happened?
What happened was Smallpox. It wiped out close to 90% of the population of the Americas in only a few decades. The cultures in the Americas didn't have the population base to copy and improve on what the Europeans were doing - in China and India, the political situation was ripe for exploit by outside powers (and this has been a recurring theme for both civilizations going back millennia), but in the Americas, it was plague and plague alone that allowed the Europeans to take over. (The Vikings had been trying off and on for 500 years before Columbus - things Did Not Go Well for them outside of Greenland.)
Well, Africa is in the Old World, but most of sub-Saharan Africa was developed less than the Maesoamerican civilizations. Pre-Christian North-East Europe was also at the stage comparable with American cultures. Siberia and North Asia were less developed also.
That is, only the European civilization developed from Classic Antiquity had significant advantage over Mesoamerican civilizations (as well as over the rest of the Old World).
Historically, civilizations have developed best along peninsulas: mostly surrounded by water, but with one land bridge. Egypt was a peninsula (between the Nile River, Red Sea, and Mediterranean). So was Babylon (between the Tigris and Euphrates Rivers). India is one large peninsula, as was ancient China (between the Yellow and Yangtze Rivers). Greece and Rome, of course, were the classic examples.
The new world had fewer such peninsulas (in good climates). You can call Patagonia, Argentina a peninsula, but that's too cold. The Panama Canal has made Mexico/Central America a "peninsula," but one that's too hot.
The WHOLE purpose of the Erie Canal was to turn the Eastern United States into a "peninsula (by connecting the Great Lakes, Atlantic Ocean and Gulf of Mexico), which is one reason why that country prospered, beginning in the nineteenth century.