On the Common Core, the educational standards first devised by a bipartisan group of governors, which have become deeply unpopular among conservative activists, Mr. Tillis also sounded far more conservative than Mr. Bush. The North Carolina House approved the standards in 2011, but, facing primary challengers from the right earlier this year, Mr. Tillis backed away from them.

“I’m not willing to settle just for a national standard if we think we can find things to set a new standard and a best practice,” Mr. Tillis said, pivoting to an attack on the federal Education Department as “a bureaucracy of 5,000 people in Washington” who make an average salary of a little more than $100,000.

While criticizing the Education Department is common among Republicans, Mr. Tillis was standing next to the younger brother of President George W. Bush, whose signature accomplishments include No Child Left Behind, the sweeping federal education law run by the department.

Mr. Bush sensed the need to play down any differences and returned to the microphone. “We can argue about what to call these things,” he said, but maintained that the focus ought to be on ensuring high standards.