You're right. DKRDS does indeed have more modes. Race, trophy race,Token Tussle, and Battle. However, the only thing I see people play is the regular races. Occasionally, I'll get a trophy race. I've yet to play Token Tussle and Battle mode. I like DKR, but I have to give my favorite racing game to MKDS. The problem in DKR is that it's very difficult to get back in first place. There aren't any advanced strategies like snaking, that could really help you get in the league. Compared to DKR, as Shiny Metagross said, all it is driving and turning. The person in first place has the same chance,or same opportunity to get a strong item just as the person in the back. Such as a purple boost(Which is the best boost you can get). Items are basically pre-determined according to their color, where as MKDS, they're randomized. MKDS at least gives you a chance to step it up. Wether it be Snaking, getting a star, lighting bolt, or bullet bill. Where as DKR, which again, Shiny Metagross pointed out, is "Who can get the best items". DKR does let you play all courses online(24 of them, compared to only getting 20 out of 32 tracks on MKDS), but I find them rather boring and somewhat repetitive. One main difference I've noticed is that DKRDS is just a remake of the original DKR. Same tracks(with 4 new ones), but just somewhat edited the tracks. Though, I have to say, the edits are nice. MKDS at least has new tracks, and brings you back some tracks from the past(SNES, N64, GBA, and GCN tracks respectively each cup). DKR has 3 vehicles to race in compared to the one kart in MKDS. but once again, none of them have advanced strategies. In fact, the most 'advanced' strategy you'll ever implement is the 'stop before you hit a zipper/boost'. The plane is a shame. All you do is turn/sharp turn in it. Car is the pretty much same thing. However, the hovercraft actually takes some practice to fully use it. But still, no real advanced strategies to help you out.
Overall, get MKDS.
And for you people who consider snaking "cheap" or "unfair", then do this: do two consecutive powersliding(without snaking). Yes, it's tougher, but when I did this, I got the same speed results as I would with snaking. Tell me, is this cheating or not?