Before the Quidditch World Cup, Mr. Crouch Sr. says:
“Oh and I’ve been wanting a word with you too, Arthur,” said Mr. Crouch, his sharp eyes falling upon Mr. Weasley. “Ali Bashir’s on the warpath. He wants a word with you about your embargo on flying carpets.”
Mr. Weasley heaved a deep sigh.
“I sent him an owl about that just last week. If I’ve told him once I’ve told him a hundred times: Carpets are defined as a Muggle Artifact by the Registry of Proscribed Charmable Objects, but will he listen?”
“Ali thinks there’s a niche in the market for a family vehicle,” said Mr. Crouch. “I remember my grandfather had an Axminster that could seat twelve — but that was before carpets were banned, of course.”
From this we can see that it was Arthur Weasley who banned carpets, because Crouch calls it "your embargo". That means this ban can not be older than Arthur, and in fact should be quite recent, as Arthur was not born into his position at the ministry. As Mr. Crouch says, he remembers that his grandfather had a (flying) carpet, he has seen it and probably was allowed to fly on it when he was younger.
On the other hand, flying brooms are obviously not banned, otherwise there would be no flying lessons and no Quidditch.
The reason Arthur gives is that they are Muggle Artifacts. But so are brooms. While the original books don't mention anything about the origin of flying brooms, Quidditch Through the Ages says that brooms were chosen because they were Muggle Artifacts.
If (wizards and witches) were to keep a means of flight in their houses, it would necessarily be something discreet, something easy to hide. The broomstick was ideal for this purpose; it required no explanation, no excuse if found by Muggles.
The same is true for a carpet, it requires no explanation, no excuse if found by Muggles.
So there is no factual reason to ban carpets but allow brooms.
One could argue cultural bias, but the fact that the ban is recent and carpets were used in England before the ban seems to invalidate that reasoning.
So why did Arthur ban carpets while he has no problem with brooms? He even created a loophole for flying cars, and nobody would deny that a car is a Muggle Artifact.
Later we hear that
Ali Bashir was caught smuggling a consignment of flying carpets into the country
If he is willing to take the risk, that means that there has to be some demand for carpets. They are not as popular as broom, but there is some demand, and they are certainly more convenient to use. What is the benefit that justifies to prohibit something a part of the population wants? Why would a charmed carpet be more dangerous than a charmed broom to Muggles?