VS:
Where it is better:
The martial arts scenes in the 2010 version are much more creative and dynamic, reflective of the use of Kung Fu, Chinese styles. Also reflective of the way western movies have come to adopt the more kinetic fighting style of Asian movie makers. The overall flow of the 2010 version improves on the lyrical pacing of the middle "training" scenes. And using the Chinese mountain range and the Great Wall as a backdrop results in breathtaking visuals. Wenwen Has as "Mei Ying" is cute as a button. And for all of us who every studied the Crane Kick in the 1984 version so closely you notice those little weak missteps (i.e., Ralph Macchio signaling to Willam Zabka which foot he's going to kick with, and the stutter step he took to time the kick properly) the finishing move in the new version is executed much better.
Where it is the same:
Both versions share the same story beats, and sometimes the exact same framing on certain scenes. Rather than being a detriment, the 2010 version seems to be playing homage to the original version rather than "stealing" or "copying". The "Pick Up The Coat" scene is a moving as the "Wax On, Wax Off" scenes. Jaden Smith not only nails the sweetness and overall goodness required for character (Ralph Macchio's specialty) he nails the more annoying traits that fully flesh out the character (there are times when both Daniel and Dre deserve to get their asses kicked).
Where it is deficient:
The 1984 version had more historical resonance by evoking the horrific consequences on the Manzar Interment Camps. The 2010 version obviously shirks any mention of politics (being one of the few western movie companies allowed to film in China) so while we can empathize with Mr. Han's personal tragedy, there's no sense of the man having to deal with fighting on behalf of a country that still viewed his race as the enemy.
Also, some movie critics wonder why Chen (the Chinese Johnny) has so much antipathy to Dre. There is a brief mention that Chen "likes" Mie Ying, but that isn't enough to justify him ready to kill the guy. Is it because American's are coming in and taking jobs from the Chinese in their own country? Is it a racial thing? The 2010 version glosses over those sort of questions.
Misc:
In the tournament sequence, Chen fights and defeats a flashy, obviously mohawked kid named "Wu Ping". Is this a reference to Master Yuen Wu Ping the fight choreographer for the Matrix and Kill Bill movies? The Master has a reputation of being dislikable. David Carradine characterized Wu Ping as a primadonna his memoir "The Kill Bill Diary". Is there some beef between Jackie Chan's stunt team and Wu Ping's stunt team?