Well DKRDS is getting more popular due to more multiplayer options (especially on wi-fi) and the fact that some people really hate snaking. I like snaking, but atm I'd have to say DKRDS is my favourite of the two. It has a really good nostalgic feel to it and there is a bit of depth with stuff like picking which upgrade and stuff. Both are real great racing games nonetheless. I feel KDS reminds me more of the SNES game, I prefer the MK64 mechanics far more than the MKDS ones. Stuff like the blocking sucks and getting hit by something or getting lightning'ed loses your items is really annoying.
Oh yeah, I forgot, on Diddy Kong Racing you can record your own sounds. That is win. I love it that every time I shoot, fps_doug yells "BOOM HEADSHOT!"