Mr. Trump, wearing a dark suit, a white shirt and a black-and-white stripe tie, entered the Clementine Hall of the palace behind a dozen attendants and was accompanied by Archbishop Georg Gänswein, prefect of the papal household.
Mr. Trump nodded with a serious face as Archbishop Gänswein, who is close to Pope Emeritus Benedict XVI, made small talk. Behind them were Mrs. Trump, wearing a black dress and with her hair covered by a veil, who listened to an attendant, and Ivanka Trump, the president’s daughter and adviser, also wearing a veil over her hair and in a black dress with lace hem and pearls.
They were followed by Ms. Trump’s husband, Jared Kushner; Secretary of State Rex W. Tillerson; and H. R. McMaster, the national security adviser, who paused to examine frescoes. With other members of the president’s retinue, they walked past seven Swiss Guards standing at attention in front of a white papal throne, halberds in hand.
The private audience ended at about 9 a.m., with the signaling of a bell, and soon after Mrs. Trump went into the pope’s studio.
The president smiled broadly as he stood next to Francis, who looked more serious but smiled as he shook Mrs. Trump’s hand. The president then introduced his daughter, Ivanka, and the rest of his delegation, beginning with Mr. Kushner.
The pope handed out rosaries to members of the delegation before the group posed for pictures, and then he bade the president and his wife farewell.
Smiles and pleasantries aside, the atmospherics of this meeting were fraught. Pope Francis and Mr. Trump have diametrically opposed views on issues as varied as immigration, climate change and arms sales. Although both men seemed determined not to let politics intrude on their encounter, the underlying tensions were clear.
On Tuesday night, Cardinal Peter Turkson, a top Vatican official with close ties to Francis, acknowledged the differences in a post on Twitter: “Pope Francis & Pres Trump reach out to Islam-world to exorcise it of rel. Violence. One offers peace of dialogue, the other security of arms,” he wrote, in an apparent reference to the $110 billion weapons sale that Mr. Trump concluded with Saudi Arabia.
The pope and the president were both elected as outsiders promising to carry the far-off voices of the forgotten to the centers of global power. But that is more or less where the similarities end.
Mr. Trump is the scion of a real estate developer and a thrice-married lover of all things gilded. Pope Francis has made a calling card out of modesty. When, in 2013, he paid his own hotel bill after being elected pope, Mr. Trump wrote on Twitter: “I don’t like seeing the Pope standing at the checkout counter (front desk) of a hotel in order to pay his bill. It’s not Pope-like!”
The two have fundamentally different views about how to restore balance to a global economic system they consider broken, with Mr. Trump focusing on the engines of capitalism and Pope Francis fighting to protect the workers and the disadvantaged from dehumanizing forces of the modern world.
“The thing they have in common is a major responsibility to govern,” said Antonio Spadaro, a Jesuit priest who edits the Vatican-approved journal La Civiltà Cattolica. “The pope is an actor on the world stage and Trump is the president of a country with a huge impact on the world.”
Pope Francis, a savvy political operator, had signaled in the days leading up to the meeting that he was not seeking a confrontation.
Speaking on the papal plane after a recent trip to Fátima, Portugal, Francis was asked what he expected from the meeting. “In our talk, things will come out, I will say what I think, he will say what he thinks, but I never, ever, wanted to make a judgment without hearing the person,” he said.
That is a far cry from his previous remarks about Mr. Trump. In February 2016, Francis responded to a question about the president’s hard line against immigrants and his desire to build a wall along the border with Mexico, telling reporters, “A person who only thinks about building walls, wherever they may be, and not building bridges, is not Christian.”
Mr. Trump, a candidate at the time, swiftly returned fire. “For a religious leader to question a person’s faith is disgraceful,” he said at a campaign rally in South Carolina.
“No leader, especially a religious leader, should have the right to question another man’s religion or faith,” he continued. “If and when the Vatican is attacked by ISIS, which as everyone knows is ISIS’ ultimate trophy, I can promise you that the pope would have only wished and prayed that Donald Trump would have been president.”