How did Hoover's and Roosevelt's approaches to handling the Great Depression differ?
Hoover trusted the markets to right themselves and Roosevelt did not.
Herbert Hoover thought of the Depression as a temporary bump that would correct itself without government interference. He saw it as largely a problem of public perception, and believed government should stat out of business affairs. When the depth of the problem became apparent to him, he attempted some fiscal reforms, but they did not have any effect.
Roosevelt, by contrast, did not trust Wall Street to fix its own problems. He instituted "bank holidays" to head of massive panic withdrawals of bank reserves. He created government-backed jobs through public works programs like the Works Projects Administration and the National Recovery Act. Conditions stabilized slowly, but the economy didn't fully recover until the US entered the war.
There's a (possibly apocryphal) story about Herbert Hoover. In the depths of the Depression, he was visited at the White House by his predecessor, Calvin Coolidge. Harding said "I don't know what to do! I've tried raising interest rates, I've tried lowering them, I've tried printing more money, nothing seems to help!" Coolidge said "When you send a bull into the cow pasture, you can't expect to see cows go to calf the next day." And Hoover replied "Yes, but you might expect to see a few contented cows!"