Good to hear that I'm not alone with a similar and somewhat unexpected difference.
 
In terms correctness (as in flat frequency response), I'd say that nothing beats measuring then. If that isn't an option, I'm squarely in the 'what-makes-your-music-more-enjoyable' corner. More of something isn't necessarily better, as balance is what you're after. On the low end, it depends mostly on if it is the very lowest frequencies as many setups (normally not transports however - in my case from the OP it's my speakers) have a rolled off bottom. Thus - extension is at least what I have found more important. Systematically go through your music and find stuff that tests particular ranges by for instance having only voices (i.e. singing 'a capella'), or having a certain instrument (or maybe two/three that aren't too close in terms of where they operate) that is central to the overall piece/song. There are good guides (google this) for which instruments cover which frequency ranges, so this can be helpful if you find one area that is different sounding than another, because what you're looking for is to nail down what subjectively sounds most natural in different frequency ranges. 
 
In regards to your dilemma - I'd easily say that the point brought up earlier in my case that the hardware implementation may be more or less successful is a probable cause.

Quote. Navyblue
 
Assuming that one sounds correct, and the other doesn't, what is this "correct" sound that I should look for? One seems to have more top end and another seems to have more bottom end in frequency response. In terms "objective" parameters, like details, frequency extensions, or harshness, I don't find one to be superior to the other, or at least not enough to tell them readily apart. What is most obvious to me is the sound signature.
 
 
 
The answer is maybe that there is no answer. At times the sound signature of something is just a personal preference. The messed up one at times is the least revealing but most enjoyable. Live with it one way for a year then change back and see what you think. I like the warmer DACs with less treble detail.
 
One of the best reads on this subject was Stereophiles review of the Digital Lens in 1996. Maybe it was placebo but the reviewer was thinking he could hear a sound signature of a transport. Entertaining in the least. I will look for it and post a link here.
 
 
 
 
http://www.stereophile.com/digitalprocessors/824/

Agreed about sound signatures and personal preference.
 
Good link Redcarmoose. Anyone interested in this topic should read it. 
 
It doesn't give a definitive answer to the OP, but does give plenty of food for thought.

redcarmoose said:

One of the best reads on this subject was Stereophiles review of the Digital Lens in 1996. Maybe it was placebo but the reviewer was thinking he could hear a sound signature of a transport. Entertaining in the least. I will look for it and post a link here.
 
http://www.stereophile.com/digitalprocessors/824/

Click to expand...

It's possible that the device could have made a perceivable difference, especially as it was adding noise in some cases above the noise floor of CDs. Don't forget, the article was published 15 years ago. These days, cheap jitter reduction circuitry in DACs mean transport or cable induced jitter is no longer a problem and the use of advanced noise-shaped dither during mastering (for a decade or so) has made the dither features of the Digital Lens redundant as well.

G