So you're routing distorted signals through two different PC interfaces.
Which audio chips are used? The Mac Mini has a fairly good Intel converter/codec (had one a few years ago), but what's in the PC?
Do you know how the headphone driver in the Alesis speakers is implemented?
It's quite possible that, while the build-in amp is ok, the headphone amp could be a simple opamp..
My experiences with my DT770 vs the old Denon AH-D300 is that the Denon is a lot easier to drive than the DT770, even this is just the 80 Ohm version.
Another maybe issue could be that the Alesis monitors could be slightly rounding-off on the tone, and when you ten plug the more linear DT770 into an opamp-based headphone amp (assuming it is so made), you'll end up with a somewhat different tonal balance.
Suggestion: If you have access to a set of those inexpensive Sony earplugs I mentioned, do give it a try.
Ahh, the model is printed on the plug: MDR-2808. Give 'em a spin!