Mr. Cohn — who wears monogrammed shirts, gold cuff links and a Rolex watch next to a brown leather bracelet with a "peace" tag and a black beaded one with silver skulls — has brought a brash style and a take-no-prisoners approach to the tax issue. He is facile with data and has a keen understanding of the economy, according to people who have attended meetings with him, and he moved quickly to hire several experienced and well-connected aides — including Jeremy Katz, his deputy, and Shahira Knight, a former tax-writing congressional aide — who have been drivers in the private talks with Congress.
But Mr. Cohn has also rankled lawmakers in both parties with comments that they suggest betray a lack of understanding about the political process and the intricate policy trade-offs that undergird a large tax rewrite. In a meeting with a group of Senate Democrats this year, according to people who were present, Mr. Cohn jokingly dismissed concerns about the wisdom and cost of repealing the estate tax, remarking, "Only morons pay the estate tax."
A source close to Mr. Cohn denied that he had used the word, saying he had been referring to "rich people with really bad tax planning."
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 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.
Mr. Cohn and Mr. Mnuchin declined to be interviewed for this story.
Aides to both deny any ill will or any attempt at one-upsmanship, describing any friction as a result of stylistic differences. And the two men have been virtually inseparable while working on the tax rewrite, talking at least once a day, dining together more than once a month at such popular Washington spots as Rasika and the Trump International Hotel, and keeping a singular focus on achieving the biggest tax cut possible in line with Mr. Trump's desires. Mr. Cohn was invited to Mr. Mnuchin's recent wedding to the actress Louise Linton, said a person who was aware of the invitation, although he did not attend.
The men have tried to avoid the mistakes that plagued the collapsed effort to repeal and replace the health care law, placing a premium on communicating with key members of Congress and finding a common approach to which the White House and Republican lawmakers could agree.
"I credit Secretary Mnuchin and Director Cohn for playing a key role in the work we have done to unify behind bold principles for tax reform, which really wasn't the case in health care," said Representative Kevin Brady of Texas, the chairman of the Ways and Means Committee.
They recently intensified their efforts to present a united front. Last week, Tony E. Sayegh Jr., a top public affairs official at Treasury, was detailed to the West Wing to work with Hope Hicks, the interim White House communications director, on messaging for the tax push.
"This is an administration where everyone is celebrating Festivus every day and airing grievances, and there's none of that here," said Rohit Kumar, a tax specialist at the accounting firm PwC and a former deputy chief of staff to Senator Mitch McConnell, the majority leader.
Still, Mr. Cohn and Mr. Mnuchin have yet to produce the kind of comprehensive plan Mr. Trump has promised. A hastily compiled one-page document they presented in April proposing an array of tax cut bullet points in different fonts and type styles drew private ridicule from lawmakers in both parties and tax specialists throughout Washington as the mark of an administration wholly unprepared for the heavy legislative lift it was about to undertake.