Ryan, any one piece of audio gear might sound different from another for many reasons. But when you ask about the sonic difference between good 16-bit and good 24-bit, that difference in itself is so slight that it wouldn't normally be noticeable unless the playback volume was set very, VERY high.
When the CD format was still in its design stage, Philips seriously proposed that it be limited to 14-bit resolution, because they had a huge stockpile of 14-bit DAC chips and for home listening, 84 dB dynamic range would already be a great improvement over the best LPs or FM broadcasts. It's considerably wider than the dynamic range of 15 ips Dolby-A master tapes. 16 bits gives another 12 dB range, which puts its noise floor far below that of any home listening environment other than a distant farmhouse in the middle of the night. Even then, if an ant scratches its groin in the pantry, you lose it.
So maybe you didn't do anything wrong at all--maybe the two things really do sound the same under the listening conditions that you have there. On the other hand, maybe that's because as some here seem to be suggesting, your setup is feeding the same 16-bit (i.e. truncated) signal to both playback systems. You should definitely check that out. A truncated 16-bit rendition of a 24-bit recording isn't valid for making any comparisons other than to test the effects of truncation itself--sooner or later you'll hear why it's bad.
The type of sonic differences described by Stagger in this thread are clearly being caused by some other, extraneous variable. It could be truncation but more likely it's something else (perhaps in addition). The only inherent difference between 16-bit and 24-bit systems is their dynamic range; there are no differences of timbre, imaging, or anything that is particular to any one part of the audio frequency spectrum. When you hear any of that stuff, it's time to take a break and figure out where your experimental design went to hell.
There is an astonishing amount of persistent nonsense floating around on this topic ... and not just on this forum by any means. When the CD was introduced in the early 1980s, the audiophile magazines went all Lou Dobbs on the infidel invader, and the simple technical facts of the situation have nearly been banished to living in caves.
--best regards