There are a lot of similarities between this remake and the 1984 original, and so the film remains incredible faithful. The only differences are the setting, the age group, different character names and the addition of some Mandarin dialogue and text).
Aside from the obvious and most central plot points (Dre runs afoul of a bully, and eventually faces him at a tournament), the following similarities are seen:
--Mr Han and Mr Miyagi are both housing maintenance men and handymen with some broken English, and see Dre/Daniel trying to learn a martial art while trying to fix the main water systems.
--Dre lives in a single-parent household, raised by his mother, much like Daniel in the original> However, Dre's father is shown to have died due to unknown reasons. Daniel's father, Mr. LaRusso is never mentioned in the original so it is unclear weather he is dead or divorced.
--Cheng and his friends are not purely malicious at heart, but are taught to be by their own martial arts instructor who treats them very badly and rewards ruthlessness. Master Li even uses the tag line used by John Kreese in the original movie: "No Mercy."
--Mr. Han tries to catch flies with chopsticks, just like Mr. Miyagi in the original.
--Dre plays a prank on his bullies by throwing a bucket of dirty, slimy, oily rainwater water at them which causes them to chase him and beat him up and having Mr. Han come to his rescue. In the original, Daniel sprays water on one of the bullies in the bathroom stall during the school dance and is also chased and beaten up with Mr. Miyagi coming to the rescue.
--Mr. Han tries to reason, make peace and part on good terms with Master Li at his kung-fu school despite their rivalry but is coldly rebuffed and rejected. Mr. Han then arranges for Dre to participate in the tournament, with Master Li making the same agreement made between John Kreese and Mr. Miyagi in the original movie: That Cheng and his fellow students will stop bullying Dre in the meanwhile.
--Mr. Han teaches Dre basics by having him do repetitive mundane house-hold related tasks. His "Jacket on, Jacket off" exercise is clearly inspired and by and based on Mr. Miyagi's famous "Wax on, Wax off" from the original movie.
--Both Mr. Han and Mr. Miyagi perform some kind of mysterious medicine healing ritual technique to a wounded, beaten up and bruised Dre and Daniel, respectively.
--Mr. Han hides a painful secret involving his family that eventually comes out while he is drunk, like Mr. Miyagi. They are both shown to be alone, with their family (wife and kids) dead.
--In the tournament semi-finals, Master Li forces his student to sacrifice his place in the tournament by inflicting a severe injury on Dre's leg, resulting in the student's disqualification. Mr. Han helps him recover enough to face Cheng in the final bout.
--During the final match, Master Li demands that Cheng focus on Dre's hurt leg, and Cheng is heitant to do such a brutal, illegal and unethical move but reluctantly complies. Dre nonetheless wins using a fancy jumping move (much more flair and pizzazz than the 'Crane kick' used by Daniel in the original movie) and finally gains the two things he needed most by the end of the movie: 1) overcoming his fears, and 2) winning the respect of his nemesis.
--Similar alternate ending scenes, involving a battle between the two martial arts instructors, were removed from both the original and this movie. In the original, an alternate ending was scrapped, where John Kreese is roughing up Johnny Lawrence for losing to Daniel at the end of the tournament. Mr. Miyagi intervenes and badly humiliates Kreese in front of all his students (this scene was later added into the beginning of The Karate Kid Part II). At the end of this movie, an alternate ending where Master Li is infuriated at Cheng and his fellow students bowing respectfully to Mr. Han, and Mr. Han fights off Master Li and leaves him sprawled on the floor, was also taken out of the final theatrical release.