Putting it simply, the sound that will be played back resides on your PC in digital format. At some point, some device will need to convert that audio into analog, and send it to your speakers. You want whatever device that produces the best analog output signal doing this process. When you connect from a PC using HDMI, you are basically having some *other* device doing this conversion. If its a $300 receiver, then you probably gain over letting the onboard audio chip do this conversion. If connecting directly to say a TV, then you lose a ton of quality.
Point is, if you have a soundcard, let it do its job. Even cheap soundcards from ASUS produce an output similar, if not better then, some $300 digital receivers these days.
Whether you can hear the difference? That depends on the speakers you connect to. But the ASUS soundcard will almost certainly produce a better output.
For me, I never recommend sound cards to anyone who isn't really an audiophile. But, if you already have it, the sound card will do better than HDMI. Like gamerk said, dedicated is pretty much always better than shared. Now, if you want to fully utilize the 7.1, HDMI might be the way to go, but you would get technically better sound quality from the sound card. How much better depends on your speakers you plan on using.