op-amps = operational amplifiers, those cute little IC (integrated circuit) packages that usually contain a series of transistors in a voltage gain (amplifier) configuration (oscillator circuits are also common).
amp = ? as a 4th year electrical engineering student i have never heard that term used to describe, what i think you are using it to describe, or contrast from an op-amp.
of the two op-amps you linked the 627 seems better on paper, faster response time, less noise/hz at listening frequencies (though both values are pretty much negligible to one another in comparison, i'm not an audiophile), but that's on paper and can vary significantly in application.
*edit* after looking at the card it looks like they let you replace the opamp/s for the headphone amplifier circuit. cool. its however uncool that the auzentech site sells their opamps for 3-4 time markup from what mouser does. for example $250 for the 627sm from them, ouch! its $80 bucks from mouser, or 45 for the 627AM which i would say with some degree of certainty you probably couldn't hear the difference (though again i'm not an audiophile) *you could also send Texas instruments an email and ask for a 'sample', never hurts to ask right?* **side note you will have to get 2 opamps for both channels in stereo but still, it's considerably cheaper . **