The big thing about the New World is that it was "new." Prior to the Age of Discovery, no one thought it existed at all. It just showed up when explorers tried to cross the Atlantic Ocean on their way to other parts of the Old World. This led to all sorts of philosophical/theological discussions, such as "Do these inhabitants have souls? How did they get from the Garden of Eden to there? If they have souls, how could God allow them to remain isolated from the True Faith for so many centuries?" [remember, we are talking ~1500]. Africa was a new area to conquer, but not open to those sorts of questions; it took until Dias for the Portuguese to reach the Cape, but no one doubted that Southern Africa existed (although there was some concern that it might extend all the way to the mythical Southern Continent and thus be impassable).
From a biological perspective, it's also not New World: the Atlantic and Pacific Oceans provide enough of a barrier to cause significant differences between the species on both sides in a way that Africa doesn't match. See for instance Old World and New World monkeys, which diverged ~40 million years ago. It also produced a disease barrier, which is one reason that Native Americans (as well as various Oceanian peoples) were absolutely devastated by disease, allowing an early, direct large-scale European conquest and settlement in a way that wasn't really replicated elsewhere for centuries.
