The #1 reason most choose and recommend ASIO: habit. Inertia.
For a long time, ASIO was a safe bet (still is) but that was before Microsoft finally got serious about audio and gave us an improved WASAPI in Windows 10. WASAPI can now rival ASIO for low-latency performance.
ASIO still has a potential edge because it's written for a specific device, and can therefore be tailored to the hardware in a way that a generic driver like WASAPI cannot. Because the interface manufacturer probably wrote the driver, you can count on it working with that interface. There have been reports of interfaces that just don't like WASAPI, but those are rare. Of course, not all ASIO drivers are created equal.
Both ASIO and WASAPI bypass the unnecessary Windows overhead (e.g. Windows mixer, "sound enhancements"), and both will
sound the same. If a detectable difference exists, it will be in efficiency - as indicated by the ability to work reliably with small buffers.
Pull up your most CPU-intensive project (as opposed to one that challenges your disk drives more than the CPU) and determine the lowest buffer size that your computer can reliably play back without dropouts. Then do the same for the other driver (bear in mind that switching drivers may also change the default buffer size, so you have to verify that you're comparing apples & apples). If both are able to handle the lowest buffer size, then flip a coin to choose your driver.