Whether one prefers the sound of ASIO over WASAPI is all a matter of personal taste. I like ASIO.
The bigger difference is in the ease of use. WASAPI is available with Windows 7 by default. So the audio device driver will be able to support WASAPI because Windows allows it to.So if the program can make use of it, it's already there.
This is not the case with ASIO. A program may be able to make use of ASIO, but the sound device driver still needs to be able to support ASIO as well. So let's say you have a sound device from company A, and company A's drivers also support ASIO... the Media Center + Company A sound device & driver is good to go.
But let's say you have a sound device from company B, and company B's driver do not support ASIO. In this case, you could use something like ASIO4ALL (which is an ASIO driver) to allow your sound device from Company B to support ASIO. So now you can use Media Center + Company B sound device + ASIO4ALL and get ASIO to work.
Does this make sense?
Also, in your last thread you mentioned about the master volume not working. Windows master volume control is not suppose to do anything with ASIO and WASAPI.