In Harry Potter and the Goblet of Fire, Mr. Weasley says that carpets are "defined as a Muggle Artefact by the Registry of Proscribed Charmable Objects," and as such were illegal in Britain. Why don't brooms fall into the same category?
It has to do with the history of broomsticks. In the days before the International Statue of Wizarding Secrecy, witches and wizards still kept to themselves, simply because they knew that Muggles would try to exploit them as well as coming to them for solutions to all their problems. As a result, the wizards and witches "hid" their magical items, including their means of flight.
The reason for using brooms is given in Quidditch Through the Ages:
As this was all before the International Statue of Wizarding Secrecy, there was no such thing as a Muggle Artefact (the whole point of restricting the enchantment of certain objects was to prevent them from falling into Muggle hands and letting out the secret or from harming the Muggle) and therefore brooms couldn't be illegal in this way.
By the time the International Statue of Secrecy did come about, brooms had been around for around 600 years, and there was no way that the Ministry was going to try to control them, especially because Quidditch was so popular:
However, the flying carpet having originated in the East, it wasn't nearly as popular at the time, and so wasn't granted this special exception.
For more on the history of brooms and Quidditch, see Quidditch Through the Ages.