Playback applications are developed using different programming languages and compiles with different versions of compilers: C, C++, Visual C++, .Net, etc. Each has different ways to access to low level computer components: processor cores, memory, PCI buses, etc. So there are lots of things which would affect sound other than the ASIO/WASAPI
My experience is ASIO sounds better than WASAPI when the computer is just running the playback application + audio driver. WASAPI sounds better when the computer is running "at least a few" something else. These "something else" can just be the port monitoring service detecting the plug-in of USB drive, smartphone, or opening a web browser.
A Tip for Windows USB audio user: Uninstall the MS Visual C++ 2010 runtime comes installed with your MS Office, and install the newer 2013 version will greatly improve sound quality. Try it yourself and you will know what I mean.