Four years ago in Denver, Bill Clinton was given the assignment of making the world believe that he liked Barack Obama and wanted him to be president. As one longtime confidant put it, “He had to go out there and say, ‘Yeah, Obama beat the blank out of me and my wife, but still, you should be with him.’ ” But that was then.

On Wednesday night here, Clinton will be tasked with a mission that has largely frustrated President Obama: Cut through the political clutter and clarify the choice in November. Explain, in his inimitable way, in language that persuadable voters in middle-class America can understand, what Obama has accomplished and why his economic policies would pull the nation out of tough times and GOP alternatives would not.

There is nothing formulaic about Clinton’s presence at the Democratic National Convention this year. He is not just another old presidential war horse being trotted out for nostalgia or a staged show of unity. When Obama called in late July to say he would be grateful if his Democratic predecessor would give the speech placing his name in nomination, something that no former commander in chief has done before, it was an acknowledgment of how much the sitting president needs the former president. And Clinton, who loves to be needed as much as he needs to be loved, responded with an enthusiasm and diligence that served as yet another signal to people close to both men that an old wound has for the most part been healed.

“He is honored that Obama asked him to do it,” said Terry McAuliffe, a former Democratic National Committee chairman. In late August, McAuliffe spent a few days with the Clintons at a beach house in East Hampton, N.Y., and said his close friend seemed obsessed with the convention assignment, continually bringing up books and quotes and ideas he was sifting through. “This speech is very important to him. He has taken the burden and put it on his shoulders.”

The convention speech, which people around Clinton say he is largely writing himself, is part of a full-scale Bill Clinton offensive that includes a series of political ads — now playing in key swing states — that feature the former president offering snippets of the themes he will expand on Wednesday. Obama’s team views this in the most positive light, noting Clinton’s talents and soaring popularity, but history shows the occasional dangers. In late May, as the Obama team was pounding away at Republican challenger Mitt Romney’s role at Bain Capital, Clinton said of the private-equity firm: “I don’t think we ought to get into a position where we say this is bad work. This is good work.” If he was all too public in his critique, it was classic Clinton as campaign manager, sending the message to the Obama team that there are ways to go after working-class voters without alienating the financial industry, a subtlety he mastered in his heyday.

The Clinton-Obama divide four years ago was political and personal. It began during the intense and at times nasty primary campaign between Obama and Hillary Rodham Clinton as intimations of racism were thrown back and forth, a sure sign of competitive overreaching involving two men (the former president was asserting himself in his wife’s campaign at that point) with strong though different bona fides on matters of race — Clinton so empathetic that he was once called the first black president, Obama on his way to becoming the real first black president. That campaign-season animosity was accentuated by diametrically disparate individual styles. Presidents 42 and 44, separated in age by 15 years on opposite ends of the baby-boom generation, have been called matter and antimatter, fire and ice, extrovert and introvert.

Clinton could spend five minutes in a Dunkin’ Donuts in Concord, N.H., and meet a stranger whose face and name and life story he could still recall two decades later. Obama spent four introspective years in New York without making a single lasting friend. He seems content to relax late at night alone in the White House residence watching ESPN’s “SportsCenter.” Clinton doesn’t like to be alone and has been known to call a pal so that they can watch a televised basketball game together hundreds of miles apart while on the telephone. Man, you see that move?

Clinton, even though once considered a boy wonder, rose the slow, traditional way, demonstrating a lifelong love of politics. He revealed his ambitions early, running for every office he could at Hot Springs High School in Arkansas until his principal declared it enough and told him he could not run for class president. He ran for secretary instead — and lost to a girlfriend. He was so bothered by that loss that he temporarily stopped talking to her. Not many things can bring Clinton to silence. The high school trend continued in college, and then into electoral politics. He took his first office, attorney general of Arkansas at age 28, reaching the governorship two years later and serving 12 years in that job, building a national reputation along the way, before running for president in 1992.

Obama in high school played basketball, smoked some pot and showed no political inclinations whatsoever. He ran for no offices then or in college at Occidental and Columbia and was 35 years old when he entered the Illinois Senate. Eight years in Springfield and two as a U.S. senator, and suddenly he was running for president. Clinton apparently considered him an upstart. In the current issue of the New Yorker, journalist Ryan Lizza quotes the late Tim Russert quoting the late Ted Kennedy quoting Clinton during the time that Obama and Hillary Clinton were competing for Kennedy’s endorsement during the 2008 primaries. “A few years ago,” the former president said to Kennedy, according to Lizza’s account, “this guy would have been carrying our bags.”

In fact, Obama did carry Bill Clinton’s bags in a metaphorical sense during the 1992 campaign. It was then, in Chicago, that Obama, just out of Harvard Law School, got his first taste of national politics by serving as the organizer of Project Vote — an effort to register voters in the inner city that tapped a needed vein of support for Clinton.

Their differences of style are far less important than matters of substance, although Clinton’s freewheeling exuberance has occasionally troubled the cautious Obama, and Obama’s seeming insularity at times has befuddled Clinton, friends of both men say. In a sense, they start and end in similar places.

On the biography side, they came from remote places (southwest Arkansas and Hawaii), rose from family dysfunction, grew up without knowing their biological fathers, and made their own way to political heights against the odds. On the political side, on most of the big issues, there is little or no space between them as pragmatic liberals, although Clinton was able to craft a clearer ideological definition as a proponent of a new Democratic “third way.” One former Clinton adviser, assessing the former president’s perspective on Obama, said: “I sense he thinks that Obama gets all the hard stuff right but doesn’t do the easy stuff at all. And it mystifies him.” Clinton, for instance, was wowed by how Obama put together a successful health-care package, something the former president failed at, yet was confounded by Obama’s inability to go around the country and sell it.

The relationship between Clinton and Obama has evolved in ­stages. The earliest step toward reconciliation might be the most telling: when Obama, as president-elect, asked Hillary Clinton to serve as secretary of state. She needed persuading, and in this case Obama and Bill Clinton were co-conspirators, both pushing the idea that she should take the job. If there were political calculations involved on either side, the simple fact that Hillary Clinton joined the Obama administration changed the dynamics; that she proved to be indefatigable and adept as Madam Secretary at once heartened her husband and deepened the appreciation of the president.

After two years during which they rarely spoke, the first public sign that Bill and Barack were teaming up on domestic policy issues came shortly after the 2010 midterm elections that proved disastrous for the Democrats, who lost control of the House and barely kept the Senate. Obama found himself making deals with House Republicans even before they took over, agreeing to some tax cuts in exchange for an extension of unemployment insurance.

On the afternoon of Dec. 11, Clinton visited the Oval Office, where he and Obama spent a long session discussing the policy and politics of the situation and how to explain the president’s position. It was, Obama said later, a “terrific conversation” — so stimulating that he thought “it might be useful” for Clinton to share his thoughts with the media and the public. Obama and Clinton seemed like tourists from Des Moines as they scrambled to find out how to open the James S. Brady Press Briefing Room in the White House and then round up enough reporters and cameras to make it worthwhile.

When the scene was finally ready, Obama entered with Clinton at his side. With a smile, he called Clinton “the other guy,” as in “I thought I’d bring the other guy in.” The word “guy” made it a term of affection and respect, marking a change from an earlier time when Obama would more often regard Clinton as simply “the other” — an unpredictable and occasionally hostile alien force. As Obama stood by, here and there making a point of his own, Clinton let loose with his pent-up insights and formulations on the economy, an issue that he said he spent at least an hour a day studying intensely, adding: “And I’m not running for anything.”

With the former president still talking, Obama soon enough excused himself and said he was under orders from his wife to leave for a Christmas party. Clinton kept at it, solo — an odd episode that prompted media speculation about a runaway ex-president plowing right over the president. It was not the first or last time that Clinton would keep talking while others went about other business, nor was it anything but what Obama wanted. His cool persona at times allows him not to worry about being upstaged. David Axelrod, his closest adviser, said it was a sign of how Obama is “centered and self-
assured.” Even so, once “the other guy” got in that old familiar room, there was no stopping him.

Early this past fall, with the campaign underway, aides to both men set up another meeting. This one was held not in the Oval Office but on the links; the two mid-handicap, mul­ligan-prone hackers talked re­election politics as they made their way around the golf course at Andrews Air Force Base.

But whatever was said that day may have been lost in translation, if not the sand traps. On Oct. 3, two weeks after the golf outing, Clinton held a reunion in Little Rock with a legion of his former campaign troops to celebrate the 20th anniversary of the day he announced his first bid for president, and much of the chatter was about how Obama and his people were struggling, not up to the old standards, unable to explain his presidency. Clinton buttonholed several former aides to make that same lament. “He talked about how you’ve got to be able to boil it down to a card that people can put in their pocket,” one former adviser recalled Clinton telling him as they sat in the Clinton Library. “You’ve got to explain your accomplishments, because it will get lost and people will not remember.”

There was always intermingling, never a clean divide between Clinton people and Obama people, and enough Clinton aides who went on to work for Obama were in Little Rock for the critique to get back to Washington. A month later, a delegation of Obama campaign advisers and pollsters led by Axelrod made a pilgrimage to Clinton’s Harlem headquarters to get his take on the campaign. They brought their polling data and computer models and laid out the information for him on Nov. 11, describing what they were learning about the voters, all the intimate details of what they intended to do, and asked for his advice. He told them to forget about attacking Romney as a flip-flopper. That would backfire, he warned, and give comfort to swing voters who wanted to dump Obama.

“They treated him like the political genius that he is, and he loved that,” one associate said. “This was great. You reach out to him and he becomes invested in the cause.”

As the contest clarified months ago, with Romney the obvious Republican choice, Clinton’s investment became more apparent. McAuliffe arranged for Clinton and Obama to appear as a tag team at a fundraiser at his sprawling home in Northern Virginia. The event on April 29 drew about 650 people and raised $3 million, but perhaps as important, it offered Clinton and Obama another opportunity to bond. At one point during the evening, ­between a larger reception and a smaller dinner for bigger donors, they spent about an hour in McAuliffe’s dining room, just the two of them and their host, talking about politics and sports. “They were on the same wavelength,” McAuliffe recalled. “You could feel the warmth.”

When they went out to face the audience, they sat on stools next to each other, two tall lefties acting like old buddies. “My job is to introduce the president,” Clinton began, in what seemed like part tutorial for Obama, part early iteration of what he would say on Wednesday. “I’m going to tell you a couple of things I hope you’ll remember and share with others. When you become president, your job is to explain where we are, say where you think we should go, have a strategy to get there and execute it. By that standard, Barack Obama deserves to be reelected president of the United States. And I’m going to tell you the only reason we’re even meeting here. I mean, this is crazy — he’s got an opponent who basically wants to do what they did before, on steroids — which will get you the same consequences we got before, on steroids.”

When the event ended, Obama and his entourage headed toward the driveway. Clinton was still in the tent, chatting with guests. “You couldn’t get him out of there,” said McAuliffe, who had to go back to tell Clinton that the president was leaving and that he might want to say goodbye.

On Wednesday night, Clinton will say hello again to his party and a vast television audience. He is supposed to get 20 minutes, including applause. The Obama team likes the odds of his making the case, if not meeting the deadline.