To me TW seems essentially modern, very existentialist, with three episodes out of twenty-six dealing with suicide and the meaning of life, or lack thereof as a main theme [*], and a bit more upfront about its philosophy, because it addresses these themes rather directly. [...] but what fascinates me about DW is how—maybe because it's a family show and has less of an inbuilt? inwritten?... meta level, if that makes any sense?—it goes straight for the epic storylines in an unselfconscious way that gives it the feeling of a modern fairy-tale or myth.
*nods a lot* I think it partly has to do with TW being an adult show, and DW a family one - DW has to have that fairy-tale wrapping, because otherwise it would be too dark. TW can address everything much more directly, and generally feels more hopeless, I think, because Jack *is* only human in his capabilities - he doesn't have the Doctor's power. And although that power can be very dangerous and dark - as WoM showed - the despair of hopelessness, and the things people will do when pushed into a corner, resonates a lot more with our modern world, I think.

(I sincerely hope that made sense. My head's a bit jumbled.)