From Nookipedia, the Animal Crossing wiki
Bunny Balloons are items given out by Phineas in the city (Animal Crossing: City Folk), given out by Redd at the Lake (River Pool) on July 4th (Animal Crossing), or given out by Players that update their model home frequently at your Happy Home Showcase in Animal Crossing: New Leaf. Unlike regular balloons, Bunny Balloons don't break if the player falls down or is shot with a slingshot while using Wi-Fi. In addition to this, they still maintain the shape of a balloon when placed inside the player's house, while other items obtained from Phineas do not.
Bunny Balloons are featured in the the Tortimer Island slingshot tours where the player must shoot down as many balloons within a set time limit. These ones are worth three points as opposed to the regular balloons, worth one.