President Obama and Republican presidential nominee Mitt Romney made clear this week that they share an overriding belief: American political and economic values should triumph in the world.

Where the two differ most is in how a debt-burdened United States, weary after more than a decade of war, should engage other nations to pursue that goal.

Their differences emerged sharply during a pair of foreign policy speeches in New York, pushing the subject of U.S. international interests and power to the center of the presidential campaign with just six weeks to go.

A proponent of American exceptionalism, Romney has consistently outlined a far tougher approach to the world than Obama has practiced. He has emphasized rewarding traditional allies such as Israel, punishing rather than cultivating difficult nations and embracing a possible military confrontation with Iran.

Obama, whom Romney has accused of “apologizing for American values,” delivered his strongest defense yet of free speech and human rights at the U.N. General Assembly on Tuesday.

But he emphasized that diplomacy and partnerships, and American assistance where wanted without heavy-handed demands from the top, remain his preferred approach to promoting those rights worldwide and dealing with antagonists such as Iran.

“It’s very clear in reading and hearing what the two candidates have to say that, at least rhetorically, there would be a significant change under President Romney,” said Karl F. Inderfurth, an assistant secretary of state in the Bill Clinton administration who is now a senior adviser at the Center for Strategic and International Studies.

Inderfurth, who is not working with either campaign, said some of the “swagger” of the George W. Bush administration would return to U.S. foreign policy under Romney.

“Obama has tried to tone that down, and he has faced pushback for doing so,” he said.

Until now, the campaign has been concerned mostly with the economy, and foreign policy has been viewed largely as a strength for the president, who was behind the killing of Osama bin Laden.

But the recent unrest in the Muslim world — revealed in the attack in Benghazi, Libya, on Sept. 11 that killed Ambassador J. Christopher Stevens and three other Americans — has exposed Obama politically and been seized upon by Romney as a product of what he calls the president’s weak engagement of the world.

The conflicting philosophies Obama and Romney outlined this week are consistent in large part with their life experiences.

Those backgrounds have given each a different vantage on the world — a former chief executive’s broad-strokes view of how it should work and a former community organizer’s details-matter assessment — and different opinions about the best way to promote U.S. interests at a time of fiscal constraint at home and rapid change abroad.

In addressing the Clinton Global Initiative on Tuesday, Romney, the former chief executive of Bain Capital, told the audience that “when I was in business, I traveled to many countries.”

He summed up the perspective he developed by declaring that “free enterprise cannot only make us better off financially, it can make us better people.”

“As the most prosperous nation in history, it is our duty to keep the engine of prosperity running — to open markets across the globe and to spread prosperity to all corners of the Earth,” Romney said. “We should do this because it’s the right moral course to help others.”

Overall, given that he has never held federal office, Romney’s record on particular foreign policy issues is thin. He also has made several missteps during the campaign — most recently, jumping into the debate over the attacks in Libya and Egypt too soon. His overarching message, however, has been consistent.

Romney has said that as president, he would speak more loudly for American interests in the world, embrace traditional allies and punish antagonists as the need arose.

He favors Israeli interests in the Jewish state’s conflict with the Palestinians, and he has spoken more martially about how he would prevent Iran from achieving a nuclear weapon.

He has pledged a firmer hand with China after years of conducting business there. He also has suggested that countries that do not share American values, including strategic nations such as Egypt, should perhaps no longer receive U.S. aid.

But Romney, beyond policy, appears most concerned with the United States’ stature in the world — its market share, in business terms.

His steady critique of the president’s foreign policy exposes their central difference: Compared with Obama, Romney believes that the nation has a far greater ability to get its way in the world.

“We feel that we are at the mercy of events, rather than shaping events,” Romney said in a tacit criticism of Obama at the Clinton Global Initiative event.

Obama spent part of his childhood in Indonesia, the son of an itinerant anthropologist. He has discussed his ability to bring an outside-in view of the United States as a result, something many abroad have welcomed in an American president.

Obama also worked as a community organizer on Chicago’s South Side, and he has brought that street-level view to his practice of foreign policy.

His approach takes into account many motivations, cultural details and history in a way that sometimes exasperates those seeking a clearer statement of U.S. interests and abilities.

Obama took office with a foreign policy designed to signal a change from what he believed were the unilateral excesses of the George W. Bush administration. Topping the list for Obama was the Iraq war.

He ended the United States’ involvement there, a move Romney criticized as abrupt, and the president is moving to do the same in Afghanistan.

In that way, Obama has cast himself as a realist, free of the “freedom agenda” ideology embraced to varying degrees by the previous administration.

Obama’s choice of force, for example, is the drone strike, which he has expanded greatly from the Bush years on similarly contested legal ground.

His use of drones has undermined his work on other issues, including his outreach to the Islamic world. That outreach has upset such traditional allies as Israel, a breach Romney has sought to turn into political capital.

Obama has tried to be evenhanded in the Israeli-Palestinian conflict, a diplomatic effort that will be largely dormant until after Election Day. He chose that approach in the belief that a neutral role is the best way for the United States to mediate the long conflict and eliminate a traditional source of anti-American anger.

But his realism has often felt arid to those trying to understand it, too calculating and lacking in passion for U.S. values and power. This is the vein Romney has sought to tap.

Obama chose his annual address to the U.N. General Assembly to deliver perhaps his most emotional defense of American values and interests in the world, using Stevens, the late ambassador, as a symbol of what he called “the best of America” — and perhaps a surrogate for Obama’s foreign policy.

Stevens was attacked amid rising protest over a YouTube video that disparaged the prophet Muhammad. Obama called the video “crude and disgusting” while defending its right to be made.

“He acted with humility, but he also stood up for a set of principles,” the president said of Stevens. “A belief that individuals should be free to determine their own destiny, and live with liberty, dignity, justice and opportunity.”