Hmm, good question (well initial question, personally, I don't care much about Neil's belief that digital has killed audio -- everything I do is digital, and it won't change, so from my standpoint, it is what it is).
Not to go against Lee and Bean, but I would have thought that it matters. Somewhere on my home computer I have the link that has been posted several times (maybe by Lee?) of various spectral analyses of a whole bunch of dither routines. It seems that the quality of dither routines matter, I would have thought the same for DAW packages.
I think Lee is right and that the packages rely on the available computer resources and native operations that are available (say, what is available inside the Mac for the DAW s/w to work with). I wouldn't imagine there is much difference in easy tasks like normalization or amplification, but compression, dither, and EQ could be a different ball of wax.
I only know enough to allow myself to sound like an idiot, but there are differences between a typical multi-band EQ and parametric EQ, and companies making EQ plugins talk about phase coherency and time coherency or whatever. Then there are essentially proprietary compression schemes like Waves L1 and Waves Ultramaximizer, etc.
I'd imagine that most of the known DAW software packages sound pretty good, but I'd also imagine there might be differences in the quality and availability of various compression and EQ effects. So if you are using say a more basic 12-band equalizer to accomplish something in one software package, someone might be getting much better results using a more advanced parametric EQ in another package. All that said, if you are using VST and AU plugins to accomplish your workflow, other than user interface, those shouldn't have a difference in sound regardless of which DAW software you use.
Who knows? I use Amadeus Pro since it is pretty cheap, runs on a Mac, does most everything I want it to, has a good feature set (say for both editing, tracking, and saving to flac, etc), seems pretty fast, and is familiar to me since I've been using it for years.