Washington (CNN) Joe Biden hasn't yet said whether he'll run for president -- but his campaign against Hillary Clinton has already quietly begun.

The vice president and the former secretary of state are already engaged in a cold war of sorts, both jockeying for the mantle of heir to President Barack Obama's legacy as they trade veiled shots over political styles, foreign policy decisions and more.

Though the two profess warm feelings toward each other, Clinton's campaign knows Biden is a threat: Polls consistently show he pulls most of his support from Clinton if he enters the race -- turning the campaign into a three-way race and giving Vermont Sen. Bernie Sanders room to pass both. Tuesday, she rolled out a long list a long list of endorsements -- more than 50 African-American mayors, many of them from the key early voting state of South Carolina.

And Biden knows he'll have to cut away at Clinton's base -- in part by wrestling Obama supporters away from her.

Biden made the contrasts hard to ignore Tuesday at a pair of tributes to former Vice President Walter Mondale, when he invoked Clinton several times.

At an evening gala, Biden even named the former secretary of state, saying she was "great" in her post as the nation's top diplomat, but couldn't carry the same weight when speaking to foreign leaders that he could.

Throughout the day, he repeatedly took shots at Clinton's remark during the first Democratic presidential debate a week ago that Republicans are among the enemies she's most proud of making.

"The other team is not the enemy," he said Tuesday night during dinner remarks at the Four Seasons hotel in Washington. "If you treat it as the enemy there is no was we can ever ever solve the problems we have to."

He added it was "naive" to believe government could work while viewing opposition parties as enemies.

Earlier, he told a crowd at George Washington University: "I really respect the members up there and I still have a lot of Republican friends. I don't think my chief enemy is the Republican Party. This is a matter of making things work."

The line has become something of a rallying cry for Biden; the vice president made similar comments at a climate change event on Monday.

"I don't consider Republicans enemies," Biden said. "They're friends."

The vice president sought to subtly one-up Clinton in several ways Tuesday. He noted that Obama offered him a choice between the jobs of vice president and secretary of state. He said he'd traveled 1.1 million miles on behalf of the United States -- topping the 1 million that Clinton has said she traveled during her four years in the Obama administration. And he said Obama tapped him, not the secretary of state, as the closer with foreign leaders.

"I will get sent to go to speak with Putin or speak to Erdogan or go speak to whomever and it's because the secretary of state -- and we've had two great secretaries of state -- but when I go they know that I'm speaking for the president," Biden said.

Clinton unveils key endorsements

But Clinton has made a number of moves to demonstrate that it'd be difficult for Biden to find a path to victory in recent days, including claiming the endorsements of more than 50 African-American mayors -- including Dennis Williams, the mayor of Wilmington, Delaware, where Biden lives.

But late Tuesday, Williams' spokeswoman said he hasn't endorsed anyone.

"Mayor Williams has yet to publicly endorse any of the current presidential candidates," Alexandra Coppadge told CNN.

"The state of Delaware is a close knit community and he would offer his support to the Vice President should he enter the race," Coppadge said.

During last week's debate, Clinton also touted as one of her biggest differences with Obama the fact that, as a woman, she'd be the first female president. On Tuesday, former Secretary of State Madeleine Albright sent a fundraising note on Clinton's behalf, saying that Clinton would "fight harder for women and girls than any occupant of the White House ever has."

The implicit message: With the backing of women and minority voters and a Southern firewall, Clinton is locking up all the support that Biden would need if he's to seriously challenge her standing as the Democratic front-runner.

Biden's bin Laden raid account changes

Perhaps most revealing of Biden's thought process -- and relevant for a Biden-versus-Clinton campaign -- is that he retold the story of the night Obama green-lit the raid in which Osama bin Laden was killed.

The public version of the story, included in Clinton's book, was that Clinton supported the raid while, she wrote, Biden "remained skeptical." Biden himself has said that in a meeting with cabinet officials, he advised the president to see if more proof of bin Laden's presence at the compound in Pakistan could be obtained.

On Tuesday, he said he reserved his final word for a private conversation after the cabinet meeting had wrapped -- when he was the last member of the administration to have Obama's ear.

"We walked out of the room and walked up stairs," Biden said. "I told him my opinion: I thought he should go, but to follow his own instincts."

The anecdote put Biden in the inner-most ranks of Obama's advisers, a closeness he sought to highlight again during his evening remarks.

He said when he accepted the vice presidential post in 2008, he insisted upon "total access" to Obama -- including access to every document that comes across Obama's desk.

And he underscored his personal friendship with the President, claiming he spends upwards of seven hours a day in Obama's presence.

His latest comments are being heavily scrutinized as the political world waits on a decision from Biden on whether he'll enter the 2016 Democratic primary, where Clinton and Sanders are the two leading candidates. A new CNN/ORC poll shows that Clinton is backed by 45% of likely Democratic primary voters nationally, with Sanders in second place at 29% and Biden in third at 18%.

A decision on whether Biden will take on Clinton is likely to come soon.

The first state filing deadlines are nine days away, which means that by delaying any further, Biden would give up chances to win delegates that would be crucial in a close nominating contest.