How did President Hoover and President Roosevelt differ in their approach towards dealing with the Great Depression?
Hoover took a hands-off approach, and Roosevelt did the opposite.
Herbert Hoover was under the impression that the stock market crash of 1929 was a simple market correction, that it would go away if everybody just acted like everything was normal, and that markets simply do these things from time to time. Billboards circa 1930 with the blurb "Wasn't the depression terrible?" kind of summed up his tone-deaf approach to massive unemployment and runs on banks. He honestly believed that government intervention was not the answer.
By the time Roosevelt took office in 1933, he understood that no quick solutions were to be had. He did start a lot of public works projects, like the Works Projects Administration (which gave a lot of people short-term employment teaching, painting post office murals, and cleaning up public lands) and the Tennessee Valley Authority (which put a lot of broke farmers to work putting a utilities infrastructure in place in parts of the South, putting the pieces of a post-agricultural economy in place).
He also instituted several "bank holidays" to discourage panic-driven depositors from taking all their money out of their banks. Austerity became the new normal in America and stayed that way until the US entered World War II.