There's a few main differences. No bananas in DS, there's coins instead which you can buy things for (including 4 new donkey kong themed stages), vehicle customization (nothing major, spoilers and exhaust pipes and such but they do alter stats).
There's some different characters, Dixie kong and tiny Kong replace pipsy and bumper if I remember correctly. No more silver coin challenges; instead there's a stupid balloon popping minigame for each track. Each boss has a third round now (except wizpig ofc) which is optional and has a gimmick like touch screen controls, you get 100 coins for winning. You can replay any boss version you want which is nice.
Powerups are different which is annoying. For example rather than missle, homing, 10 missles it's missle, 5 missles, homing. There is a cool thing though that in hard to reach parts of the level there's these tokens you can hold onto one at a time that mega power up items to a 4th stage (missle that hits 1st place or fake coins which are bombs for green balloons for example).
Drumstick can be unlocked from the very beginning as you fling a frog onto a cliff to give him feathers using the touch screen. TT is actually realistic to unlock now, same method, but the ghost times are easier. Beating adventure one unlocks taj and beating adventure two unlocks Wizpig but it's weird because they're both shrunk down to fit in a car. Also weird because you can race wizpig as wizpig and when playing as taj he says "I must be losing it, I'm talking to myself!" Both their stats are identical, they're basically broken and overpowered as fuck in terms of speed but supposedly handle like shit (I don't mind it actually you can corner pretty well).
Also the battle mode is completely gone, the TT amulet is replaced by 4 different races against TT that are easy as fuck.
Basically all I can think of.
