In Dragon Age: Origins, I would say yes, a mage, rogue, and warrior each merit a playthrough as they use very different styles, so you will have a very different experience. And I suppose if you play your RPGs with serious role-playing, then you can do the same in Skyrim... But for me Skyrim offered the opportunity to not just be a jack-of-all-trades, but to excel at practically all of them, so I would never have various games for various character types, as I always try to use a mixture of everything.

As for race, from what I understand it doesn't really play a large factor in anything, aside from maybe a few bits of dialogue? I don't know for certain, but unless there are substantial differences between races I wouldn't even consider starting another game just to try them out. But to each their own.