While the two men are the President's leading emissaries to Capitol Hill on taxes, several Republican members of Congress and aides involved in the tax reform effort tell CNN that Cohn is viewed as the stronger advocate -- an attribute that hasn't been lost on top White House aides, including chief of staff John Kelly.
Mnuchin brings with him the weight of the Treasury Department -- a pivotal resource in the tax push -- but he has rubbed several lawmakers the wrong way with what they perceive as a blend of awkwardness, cockiness and clumsy salesmanship, according to Republicans involved in the process.
His relationship with Trump is strained, people close to both men tell CNN, but he has built a strong rapport with some members on Capitol Hill who view him as a competent partner as tax reform moves to the top of the agenda. Sources say Cohn is expected to stay in his post at least through the administration's tax push.
"He is just a heavyweight in these meetings," one Republican familiar with the administration's tax reform push said of Cohn, adding that he is particularly well-regarded because members can always count on his "candor."
Mnuchin, by contrast, is often viewed as a company man -- regurgitating talking points even in private discussions. Both men are longtime Democrats, with deep experience on Wall Street, who have spent little time in Washington or dealing closely with conservatives.
The White House offered support for both men Wednesday, praising their Hill outreach in particular.
"Secretary Mnuchin and Director Cohn are regularly talking with and listening to members of Congress on important economic issues," a White House spokesperson said. "They freely share ideas on ways to create jobs and grow our economy, and both look forward to a productive relationship as they all work together to deliver tax reform to the American people."
Mnuchin managed to infuriate several members in a closed-door meeting with House Republicans late last week when he asked them to support a three-month spending plan and debt ceiling increase "for me," essentially as a favor. The reaction ranged from outrage to disbelief -- both on Capitol Hill and within the White House.
At the time, Rep. David Brat, a Virginia Republican, called Mnuchin's arguments "intellectually insulting."
A Republican close to the tax-reform effort said Mnuchin has had several similar awkward moments during private meetings on tax reform.
It didn't help his standing when Trump abruptly undercut his own treasury secretary at a recent meeting of bipartisan leaders to plot a path forward on fiscal deadlines and disaster relief.
As Mnuchin was in the midst of his pitch for a lengthy debt ceiling increase to inject a dose of calm and stability to credit markets, Trump cut him off and made clear he was siding with the Democrats in the room, Chuck Schumer and Nancy Pelosi, according to a source familiar with the meeting.
Political operatives and donors with close ties to the White House have also been skeptical of Mnuchin's ability to shepherd legislation as complicated and politically perilous as tax reform.
At a recent donor retreat for House Speaker Paul Ryan in Jackson Hole, Wyoming, Mnuchin appeared on a panel, along with a top tax policy staffer from Ryan's office, to discuss overhauling the tax code.
One attendee described Mnuchin's performance as "babble" and "general pabulum."
The source said it was jarring to see the treasury secretary appear alongside a House staffer. "It didn't elevate him, let me put it that way," the source said of Mnuchin's appearance.
The staffer from Ryan's office fielded many of the most detailed queries. After the panel wrapped up, people rushed to exchange contact information -- with the staffer.
"Secretary Mnuchin has played a key role throughout this years tax reform discussions between the House, Senate and Trump administration. At Speaker Ryan's summer retreat, the secretary's presentation was well received and informative for our attendees," said Kevin Seifert, the executive director of Team Ryan.
A source familiar with the event added that it's not unusual to pair a high-profile guest, such as a Cabinet secretary, with a policy staffer.
Still, some lawmakers expressed optimism that Mnuchin would approach the tax reform pitch with more finesse than he brought to the debt deal.
Rep. Mark Walker, a leading House conservative who was critical of Mnuchin's pitch on the debt ceiling, sounded like he was willing to give the secretary another chance.
Walker said tax reform is "a little bit more in his fieldhouse if you will, just speaking on tax reform than maybe understanding historically why a debt ceiling -- clean debt ceiling -- were problems for conservatives, so I'm looking at that a little bit more hopeful, but reserve final comment to see how he performs in this particular area."
Rep. Kevin Brady, chairman of the House Ways and Means Committee and one of so-called Big Six tax negotiators, defended the administration's efforts so far. On Wednesday, he set a timeline to reveal a tax reform framework the week of September 25 and tackle this heavy legislative lift by the end of the year.
"They've been terrific on tax reform and terrific in the working group," Brady said of Cohn and Mnuchin. "They bring policy expertise to the table within their agencies, they bring real life expertise to the table and they are determined like we are to deliver pro-growth tax reform this year. I've got nothing but good things to say."
Dave Camp, a former Republican congressman from Michigan who spent years working on tax reform, praised the White House for placing a priority on tax reform and for spending months working with the House and Senate on a plan. He downplayed any criticism of Mnuchin's efforts.
"I can't recall a treasury secretary that has been a grassroots Republican as members of Congress often are," Camp said in an interview. "I don't think it's a drawback, necessarily. Tax reform is a challenging topic for anyone at any time."