The first thing they've changed are the controls. You use the control stick for left and right movement, and A button to jump.
They've added the ability to handslap the ground, getting bananas in some places. They've also added big seeds in some places that are underground, and that must be handslapped a few times in order to get the bananas/life/heart out of it.
Speaking of which, they've added extra life coins (which are DK coins!) and hearts that act as health. Now, instead of losing bananas when you get hurt, you lose a heart.
Another new ability they've added is a super high back flip/jump. This is useful for getting high up items, and it also gives you a combo. Also, now you can instantly slam to the ground from the air.
They've redone all the enemies a little bit, the most notable of which is the (former) Panda bear. It has now been changed to a brown bear the first time you see him, and a blue bear the second time you see him. They've mixed up the enemy placement too, and added more of them in. There's also a new electricity enemy, apparently from the original Donkey Kong Jr.!
There are now logs that will appear from time to time that you can smash for bananas, as well as cacti(but these hurt you if you touch them).
The Kingdoms (and which levels are in which kingdoms) have been mixed up a little.
In the original, when you beat a level you got a bronze crest, and if you had a certain number of bananas you could get the silver, gold, and then platinum ones. In this game, you get a Crown for beating it, and then you get 1 crest, and then a second crest, and then a third crest. They are all just crests, and are no longer named bronze-platinum. Also, I don't know the numbers of the original, but I do remember that it took 1200 bananas to get the Platinum crest. In this one it only takes 1000 to get the third crest.
This game has a story. The story is that Ghastly King is wreaking havoc on the jungle kingdoms and has taken all their bananas. DK has to help them.
They've added in a new variety of Party Monkey that sits in a brown bush holding a sign, waiting for you to talk to him. He has glasses and gives you advice about the game or progresses the story line. I like to call him Smarty Monkey.
Another new variety of Party Monkey is one that sits in a Blue bush. Once you clap at him he raises a DK flag. This is a level checkpoint, so if you die, you come back to this guy.
I don't remember if the original had this, but now Ghastly King shoots out balls of energy that are difficult to avoid.
Oh, and they've added Barrel Cannons.
Also, now whenever you get 200, 500, 1000, and 5000 bananas you get a prize. If you have full health it's an extra life, and if you're hurt it's a heart.
Now those flower blooms that you can collect will give you a life if you get 100.
Now the little bubble things that but DK into a bubble and make him float upward are controlled differently. Now you hold the Wii mote straight up and tilt it side to side to roll the bubble.
Another new addition is a new Kingdom that comes once you've gotten every crest in every other Kingdom. It's called Kong of the Mountain. This is a level in which you fight all four evil Kongs and then Ghastly King in a row with only 500 health. Once you beat Ghastly King you receive 800 bananas, meaning that you must keep at least 200 bananas the whole time in order to get all the crests(which is harder than it sounds).
After you've beaten Kong of the Mountain and gotten all the crests, you unlock one final level, called Banana Banquet. This is basically just like the ending ceremony, only now there are a bunch of bananas for you to collect. After you get all three crests in this level you've beaten the game!