No, you don't need to play ToS to play ToS-D, but if you have any intention of picking up ToS again (which you should, you didn't even get anywhere in the game...) and want it to be a fresh experience, you should hold off on DotNW because it's pretty much going to throw every major plot point and spoiler in your face at one point or another.
Symphonia is a more complete package because it's a mothership title, but I think DotNW addressed a lot of the things I felt were off about the first, like voicing skits this time, a more refined battle system, a few more of the Tales traditions and less tedious dungeons. There are also some other pluses, like more humor in general and the monster recruitment system. I'm actually spending more time with DotNW (on my second playthrough) than I did with Symphonia because of the monsters.