Mr. Mnuchin, bespectacled and crisply formal in his slim-cut business suits, is more reserved and careful than Mr. Cohn. A quick study on economic policy matters, he talks frequently about his level of access to the president and the strength of their relationship, making it clear to members of Congress as well as tax lobbyists that he has Mr. Trump’s ear and is speaking for him on the tax plan. Mr. Mnuchin has been deferential to lawmakers, indulging House Republicans for weeks in talks about their idea of imposing a 20 percent border adjustment tax even after it became clear that the idea did not have enough support to be included in the plan.

“Gary is pretty direct — he’s come to the whip team meeting and he’s probably less politically correct, and I use that in a loving way,” said Representative Pat Tiberi, Republican of Ohio and a member of the Ways and Means Committee. “The secretary is more correct. They complement one another.”

Early in their respective roles on Trump’s team, the two appeared to be jockeying for position in what one person who observed the dynamics described as an “unholy alliance.”

During the transition, Mr. Mnuchin made a surprise visit to Mr. Cohn’s job interview with Mr. Trump, greeting Mr. Cohn in the 26th-floor lobby of Trump Tower in Manhattan to escort him to the president-elect.

In meetings early this year before Mr. Mnuchin was confirmed, Mr. Cohn told key lawmakers that he would be leading the tax effort for the administration as the chief architect of Mr. Trump’s economic policies, according to several people familiar with the conversations who spoke about them on the condition of anonymity. Mr. Mnuchin appeared “zen” when he learned of the meetings, said one person who was aware of the tension, assuring staff members that Mr. Cohn, whose job does not require Senate confirmation, was doing the right thing.

“They are working together,” Mr. Moore said of Mr. Cohn and Mr. Mnuchin. “But there’s no question they’re competitors.”Their differences extend beyond style or policy. After the racially charged violence at a neo-Nazi and white supremacist march this month in Charlottesville, Va., the two men, both of whom are Jewish, had vastly different reactions to the president’s equivocating response. Mr. Cohn condemned the administration’s approach in a newspaper interview, letting it be known that he considered resigning over the issue, while Mr. Mnuchin vigorously defended Mr. Trump, saying he never dreamed of quitting.