He is Special

This part is dedicated to the power, abilities and weapons that Bunnymund uses in his everyday life and combat.

Chocolatier. Ninja. Zen Master
He lives on Easter Island. He makes chocolate. He can travel the galaxy before you can say "egg". Yes he's the Easter bunny, but he's not like any bunny you've ever seen. When he's not creating most delicious confections on the planet, he's helping the Guardians fight the Nightmare King. And with his Warrior Eggs armed and ready, E. Aster Bunnymund may be just what the Guardians need to put an end to the battle for the children of Earth
- Guardians of Childhood Official site

1. Rabbit Prowess
This means that he can run and jump like a rabbit, which he is. On land he is the fastest of the Guardians, when it comes to run and leap around. His speed and jumping are what makes him also different from others.

2. Power over Flora
Remember the flower that appears when his tunnel closes, that's one of the powers that come with the flora. Mostly this power is noticed in the Warren, where he has all those special plants that make eggs, coloring them and similar. It's also said that he can feel thru earth, so I presume that's also part of his power.

3. Rabbit Tunnels
It's his way to travel the world. His tunnels are all around the planet, and he can jump out of them wherever he wants. Aside his Rabbit prowess, this is another thing that aids him to be the fastest.

4. Martial arts
In the movie at some point he claims to be master of Tai Chi. In the book on the other hand it's says not only that he is a ninja and master at the martial arts, but that he even invented some of them.

5. Magic
Well, he has to have some magic, he uses enchanted boomerangs and explosive eggs in combat. While on the other side, the Bunnymund from books does not use a weapon, but magic - he eats a chocolate egg made by him (and recipe is also his) gaining mysterious powers (I shall not comment on this)

6. Boomerangs and Explosive eggs
This is Bunnymund's attire for combat in the movie. We see him welding this really nicely made boomerangs, that come always to his paws, even when he is not looking (Enchanted I say). They do not look as normal boomerangs, not because they are enchanted, but if you look closely, you'll see that they are a little less curved, and I'm almost sure if he put them together they would made an egg shape. They are also made of wood, because he's the nature. And he is also seen carrying and using those cute egg bombs. He somehow looks like Crocodile Dundee when he goes in combat.
Why were the Boomerangs chosen, even if in the books Aster carries a staff (that he doesn't use in combat)? Well, let's look it from this, while in the book he does not live under Australia, even if it's his favorite continent, the Bunnymund of the movie lives under the Aussie continent, which in the centuries or how much he lived there, he got the slang and changed his way of combat, because he probably liked that kind of weapon (aside you say boomerang and everyone thinks of Australia), he somehow looks like and Australian Ranger - wait I read something about it... ah yes

“In early designs, Bunny was professorial, wore a cloak, and was a little uptight. But then when Hugh was cast to play Bunny, the artists came up with this notion of him as a cool and tough Australian ranger.”— Rise of the Guardians Producer, Christina Steinberg.