Nude photos of the third in line to the throne partying in a Las Vegas hotel were published by US website TMZ earlier in the week, and then picked up by several other publications.
However, British newspapers did not run the photos after St James's Palace sent out legal letters warning them that doing so would be an invasion of the Prince's privacy.
David Dinsmore, managing editor of The Sun, said that the paper had thought "long and hard" about publication, but judged that the case is "about the freedom of the press".
He added: "This is about the ludicrous situation where a picture can be seen by hundreds of millions of people around the world on the internet but can't be seen in the nation's favourite paper read by eight million people every day.
"This is about our readers getting involved in the discussion with the man who is third in line to the throne - it's as simple as that."
With the headline "Heir it is!", The Sun has today published the photos on its front page, along with an editorial explaining the reasoning behind the decision to print.
It reads: "The photos have potential implications for the Prince's image representing Britain around the world.
"There are questions over his security during the Las Vegas holiday. Questions as to whether his position in the Army might be affected. Further, we believe Harry has compromised his own privacy."
The editorial added that publishing the photos was "a crucial test of Britain's free Press".
"It is absurd that in the internet age newspapers like The Sun could be stopped from publishing stories and pictures already seen by millions on the free-for-all that is the web," the paper added.
Copyright: Rex Features Paul Grover/Rex FeaturesThe Palace had contacted the Press Complaints Commission (PCC) on Wednesday because it was concerned that the 27-year-old Prince's privacy could be intruded upon. At that stage no newspaper had run the images.
In reaction to The Sun's decision, a palace spokesman said today: "We have made our views on Prince Harry's privacy known. Newspapers regulate themselves, so the publication of the photographs is ultimately a decision for editors to make.
"We have no further comment to make either on the publication of the photographs or on the story itself concerning Prince Harry's private holiday in Las Vegas."
The PCC said that it would investigate the matter "following normal procedures" if it received any complaints.
Earlier in the week, a former executive editor of the News of the World said that the decision by UK papers not to immediately run the images showed that they had been "neutered" by the Leveson Inquiry into press ethics and standards, which was set up after the phone hacking scandal.
However, ex-Deputy Prime Minister John Prescott said that The Sun had shown "absolute utter contempt" for the law and the Leveson Inquiry.
And Commons culture, media and sport select committee chairman John Whittingdale said of The Sun's decision: "The fact that [the photos] happened is well known. How the public interest is served by doing this is not clear."
Press standards campaign group Hacked Off also criticised The Sun's decision. Its director Brian Cathcart told the Standard: "This is the country's biggest-selling newspaper breaking the industry's own code despite clear warnings. It is flagrant proof that our national newspapers are incapable of regulating their own affairs.
"The Sun's argument that this is about freedom of the press is nonsense. This is about The Sun's right to trample over the industry's own feeble rules when it likes, and also to invade people's privacy whenever it chooses."
Mensch, who is stepping down as an MP, told BBC Radio 4's Today programme: "I'm chilled to hear the Press Complaints Commission (PCC) tried to tread on this story and they should not have done it.
"The PCC after the Leveson Inquiry really ought not to be in the business of collectively telling newspaper editors they can't run a story and they shouldn't use their best judgment. That is the real sin in this story and The Sun was right to publish."
Former Sun editor Kelvin Mackenzie said that it was important for newspapers to "start pushing back" against the "establishment".
"I'm unsure why the establishment hate newspapers so much but what I'd like to see is editors get off their knees and start pushing back against these curtailments in what will eventually, I promise you, lead to the closure of newspapers," he said.
"People should stop worrying about privacy and start worrying about what free speech will mean to this country if the Levesons and the Camerons of this country have their way."
Speaking on Daybreak, Clifford said: "We're the only country in the world where you can't read about it in the papers. It's a bit of a ludicrous situation.
"Now we'll have to see what the Palace will do. You can only assume that the Palace are going to sue The Sun. It will be interesting to see how it develops."
Clifford said that there was a "huge public interest" in the story, although he added that it was unclear whether the paper should have respected the Prince's privacy.
"I think [Prince Harry has] done us proud [with his charity work and military service]. I think we should have respected his privacy," he told the ITV1 programme.
"I understand why The Sun has done it - they're going to sell a lot more papers. But does that make it right?"