By the time Mr. Bush left office, he had shaved off some of the more controversial edges of his counterterrorism program, both because of pressure from Congress and the courts and because he wanted to leave behind policies that would endure. He had closed the secret C.I.A. prisons, obtained Congressional approval for warrantless surveillance and military commissions, and worked to close the prison at Guantánamo Bay, Cuba.

So while Mr. Obama banned harsh interrogation techniques, he preserved much of what he inherited, with some additional safeguards; expanded Mr. Bush’s drone campaign; and kept on veterans of the antiterrorism wars like Mr. Brennan. Some efforts at change were thwarted, like his vow to close the Guantánamo prison and to try Sept. 11 plotters in civilian court.

“These are the same issues we’ve been grappling with for years that are uncomfortable given our legal structures and the nature of the threat, but the Obama team is addressing these issues the same way we did,” said Juan Carlos Zarate, who was Mr. Bush’s deputy national security adviser for combating terrorism.

Peter D. Feaver, a Duke University professor and former Bush national security aide, said Mr. Obama “believed the cartoon version of the Bush critique so that Bush wasn’t just trying to make tough calls how to protect America in conditions of uncertainty, Bush actually was trying to grab power for nefarious purposes.”