i own both and i highly prefer the Wii version, but i'm not a kid
it's the analog control that does it for me. the DS is digital steering, and simply is not nearly as precise. also, MKW allows like four control methods -- just Wiimote (with or without plastic wheel), Wiimote+nunchuck (best IMO), Wii Classic controller, or GameCube controller
full disclosure: i won my DSi & MKDS in a MKW tournament, so perhaps i'm biased due to that =)
I think probably Mario Kart DS because despite that Mario Kart Wii is a pretty good game, the 150cc mode is punishingly difficult due to the rubber-band A.I. and the power-up system being more unbalanced. When you are in first place, the power-ups you get are pathetic, while the opponents behind you are treated to a luxury of stronger power-ups that mostly focus on slowing you down! Not all the other Mario Kart titles are like that.
I have both, & it would be hard to say which is better ...they have different tracks & features. control is tighter on the Wii, but some of the tracks on the DS are better ... I had it for the Wii first, & liked it so much I wanted a portable version as well... glad I have both.