Unless you have great hardware, and great hearing, there's not much difference. The 'nice' thing with WASAPI for most users is that it gives exclusive control of an audio device to the source of the sound, so you don't have your music broken up by system sounds. This alone is awesome and worth using WASAPI for.
WASAPI also gives you 'bit perfect playback'. This means it should not be altered in anyway throughout the stream - so the sound you get, in theory shouldn't be resampled or otherwise altered in any way. Practically, you'll may be running some plugins anyway so the usefulness of this is limited unless you're sending the audio stream to another device.
Practically I've messed with ASIO (tricky), WASAPI and Direct Audio on both the external DAW or DAC I use for proper listening and nice headphones and my internal soundcard in various configurations. I find that being able to allocate the external sound device to my music player alone is worth using WSAPI. I subjectively prefer the sound quality with WASAPI (and ASIO on an old device that supported only that), but the objective convinience of having nothing but my audio stream to a device is the main factor here.
I'd say try it out, and see what you like better. Your speakers/headphones and audio device are going to make a bigger difference than what output you use.