They are definitely too different to make a simple, universally applicable comparison, however people can still have an opinion. For example, I prefer oranges to apples, but there is no objective way to show that oranges are better.
Anyway, I've got both (paid for minecraft) and I enjoy them both, but I have to say I think DF is a much more satisfying and enjoyable game overall. In single player mode DF wins hands-down. Frankly I can't think of a single aspect of single player minecraft I prefer over DF, including the graphics since 3D graphics on that scale lag my laptop.
Multiplayer is the one redeeming feature of minecraft, but the community more than ruins that. Constant cries of "OMG GREEFERZ GONNA BLOW UP MAH CITIEZ!!!" all over the forums/in-game chat just get tiresome. I've found a couple of good free-for-all PVP servers, but even those get boring after a while. I've been playing minecraft during exams because it's easier to only play for an hour or two then quit, but now that exams are over I think I'll be playing DF 95% of the time. Fortunately the learning curve of DF keeps the more immature players out (read: 90% of the MC community).
Oh, and minecraft also costs money, and hence the developers are running it for profit rather for the players (yes, there is a huge difference). I plan on donating considerably more to Toady than I paid for minecraft.