ASIO's initial popularity was mainly because it allowed you to bypass Windows' built-in mixer. When Microsoft designed its intrinsic audio support, it wasn't thinking about music production, but rather the other 99.99% of applications dubbed "multimedia". Which was perfectly reasonable. Nobody cares about a few milliseconds of added latency when playing video games, showing corporate slide shows or watching porn.
ASIO came along to fill a need for an audio subsystem that incurred as little overhead as possible. As home recording and Windows-based audio production took off in popularity, the number of users grew to a point where Microsoft could justify revamping their audio support to accommodate power users. WASAPI was their answer to ASIO. It's nearly as efficient as ASIO but far more sophisticated. It can be either exclusive like ASIO or shared like DirectX.
So why is ASIO still around? Because WASAPI is Windows-exclusive. It's much easier for vendors to write ASIO drivers for both PC and Mac platforms in one go. When I had a MOTU interface, it was obvious that the company's background is Mac- and ASIO-oriented. That's when I started using ASIO, because MOTU's ASIO driver was better than their WDM/KS driver. ASIO works great with my current Focusrite product, so I'm sticking with it (ain't broke). It's likely that some interfaces still do better with ASIO because those companies have a longer history with it.
ASIO4All, btw, is not really an ASIO driver. It's WDM under the hood. Its purpose is to provide a faux-ASIO driver for interfaces that don't provide one of their own. Should be a last-resort option when nothing else works. (Correction: second-to-last option; MME always works.)