He warned against the special forces raid into Pakistan that resulted in the death of Osama bin Laden, urging that they wait for more intelligence confirming his presence, while she recommended going forward with the operation. She was a prime advocate of arming and training opposition forces in Syria, a push colleagues did not remember him joining. In recent days, she came out against Mr. Obama’s trade pact with Pacific nations, while the vice president endorsed it.

Put together, the disagreements underscore a broader philosophical schism over America’s role in the world a dozen years after the invasion of Iraq.

“He may be more cautious about the outcomes of the significant use of military force,” said Barry Pavel, a national security official in the White House during Mr. Obama’s first term and now a vice president of the Atlantic Council, a Washington think tank. “She may be more robust in ensuring that U.S. engagement is felt in a meaningful way and there isn’t a perception of a U.S. withdrawal or disengagement.”