How did Roosevelt's New Deal differ from the actions taken by Hoover following the Stock Market crash of 1929?
The main difference between the actions of FDR and those of Hoover was that FDR was willing to be bolder than Hoover. He was willing to try more things on a bigger scale than Hoover had been. As the "Salem on History" link below says,
...many of the concepts held by Hoover became part of the New Deal. Franklin D. Roosevelt, however, carried many ideas further and faster than Hoover could have tolerated.
Hoover has a very bad image in history. It is said that he did nothing to try to end the Depression. However, this is not true. He did much more than any previous president had done in terms of using government power to try to improve the economy. He set up a program where the government loaned money to banks in hopes that they would loan to businesses and the businesses would not fire workers (the RFC). He engaged in major public works projects (the Hoover Dam, for example).
Although Hoover was active, he was not nearly as active as FDR. FDR's New Deal did everything that Hoover had tried and more. FDR was willing to engage in many more kinds of actions (the NRA, the CCC, etc) to get the economy going. He was also willing to spend a great deal more money than Hoover was.
So the real difference is more in quantity than in quality. Hoover and FDR had many of the same ideas, but FDR was willing to be bolder and to move faster than Hoover was.