The campaign's social media manager Mike Hahn posted a screenshot of a message from the company, which said the post had "violated our rules against posting private information".
Twitter also locked the accounts of White House Press Secretary Kayleigh McEnany and One America News Network host Jack Posobiec for sharing the New York Post article.
Later, CEO Jack Dorsey admitted the company had failed to sufficiently explain why it had stepped in.
"We want to provide some much-needed clarity around the actions we've taken," Twitter said in a statement.
"The images contained in the article included personal and private information – like emails and phone numbers – which violate our rules.
"We also currently view materials included in the articles as violations of our Hacked Materials Policy.
"The policy prohibits the use of our service to distribute content obtained without authorisation. We don't want to incentivise hacking by allowing Twitter to be used as distribution for possibly illegally obtained materials.
"We know we have more work to do to provide clarity in our product when we enforce our rules in this manner."
That explanation did nothing to satisfy Twitter's critics, who accused it of interfering to help Biden politically.
"It's out of control. And it's like a third arm – maybe a first arm – of the DNC [Democratic National Committee]," Trump told Fox Business in an interview.
"Twitter, Facebook, they're all – like, really, it's a massive campaign contribution. This is a third arm of the DNC, the radical left movement," Trump concluded.
The Trump campaign's Twitter account has since been unlocked and the video in question has been reposted.
Republicans in Congress, meanwhile, demanded Twitter's Jack Dorsey and Facebook's Mark Zuckerberg appear at a Senate committee hearing on "digital platforms and election interference" at some point before the election on November 3.
"As your company is no doubt aware, corporations are forbidden from contributing anything of value – financial or otherwise – to support the election of candidates for public office," Senator Josh Hawley wrote in a letter to Dorsey.
"This hearing will consider potential campaign law violations arising from your company's decision, on October 14, to support the presidential campaign of Joe Biden by asymmetrically applying its terms of service and restricting the distribution of a New York Post article, as well as by suspending the official account of the presidential campaign of Donald Trump for discussing this story."
Senator Deb Fischer said Twitter's actions were "ridiculous and unacceptable".
Trump and Biden are both holding town hall events today, in the time slot that would have been used for the second presidential debate if it had not been cancelled.
The New York Post article and the conduct of social media companies are both likely to come up at some point.
