You are no doubt aware that Grant was President long before World War I. I assume you mean post war corruption under Harding, and how it compared to Grant.
Historians tend to agree that both Presidents were basically honest men with some personal flaws, primarily their tendency to trust those who could not be trusted. Warren Harding once famously said, I can deal with my enemies; it's my friends that keep me walking the floor at night."
The major scandal of Grant's administration was the Credit Mobilier Scandal, involving construction of the Trans-Continental Railroad. The company, a French company which had built railroads in Europe, charged exorbitant rates for labor and supplies to the government, and then bribed or gave stock to Congressmen and other government officials to cover its acts. Grant's Vice President, Schyler Colfax, was among those implicated. Another problem was the Whiskey Ring, in which bribes were paid to bilk the government out of taxes on whiskey. Grant's personal secretary was implicated. William Belknap, Grant's Secretary of War, was impeached for selling franchises for Indian trading posts; but resigned before he was convicted. On leaving office, Grant apologized to the American people for his missteps.
Warren Harding also trusted people he shouldn't. The most telling scandal of his administration was the Teapot Dome scandal, in which his Secretary of the Interior, Albert B. Fall, made a deal for the sale of Naval Oil reserves to Oil interests with whom he was connected and by which he profited handsomely. Also, an official of the Veterans Administration stole medical supplies with abandon. There was influence peddling in the Justice Department for which no punishment was ever administered. Harding's Chief Counsel committed suicide in Harding's old Washington residence; the influence peddler also shot himself. His Attorney General was implicated in the fraudulent sale of German assets seized during World War I. He took the 5th Amendment when called to testify; and ultimately was not charged due to lack of evidence. The lack of evidence was because he had destroyed most of it.
Harding did not live to apologize. Overcome with pressures from the scandals, he and his wife left for a vacation to Alaska, but en route he died of a stroke or heart attack in San Francisco.