There are roughly 4 basic parts to a CD player that help determine sound quality:
1. Transport-- how well does it read the data? How stable is the mechanism, how much jitter is generated, yada yada.
2. DAC set/digital board including clock etc. There are few manufacturs of DACs, all CD makers draw from this small lot of producers, so differences in good-quality digital sections get smaller and smaller all the time (although you certainly do get more and better by paying more). Many computer sound cards will have impressive specs on paper because they use the same parts, or parts with similar capabilities as stand-alone CDP.
3. Analog output section. You may have a state of the art 24/192 DAC set but if hands that off to a noisy, feeble analog section, what's the point? Do not undersestimate this!
4. Power supply. Good, stable, robust clean power is crucial to good audio performance.
IMO, PC soundcards have two major disadvantages:
1. Namely a power supply designed to power a computer, not play back music in an audiophile manner. The power supply will be noisy, cheap, plus there are a lot of noise generating components in PC that aren't in a stand-alone CDP adding to the problem. This is actually a big deal, and one of the major things you get from climbing up the audiophile CDP ladder is a better and better power supply section.
2. Analog output section, which will be cheap, typically on the same board as the DAC, and a total afterthought, probably both sharing the same noisy power supply.
IMO, with these inherent built-in disadvantages, there is no sound card on Earth (no I haven't heard them all) that could touch a well-chosen, new $1200 CDP.
BTW: DAC sets have come a LONG way in the last 5 years. If you had another $1200-$1600 to invest in a new CDP, I bet it would be a significant upgrade over your Anthem.
OTOH, if you are using the computer's digital output into an outboard DAC (like the Bel Canto), the need for a good analog section is taken care of by the DAC of course. No idea how a computer compares to a decent CDP as a digital transport to an out-board DAC, but I'd bet dollars to donuts, the CDP will be better in that function.
Mark