Was the New Deal really new? How is FDR's policy different from that of Hoover? And why was this a monumental shift in what democratic government was all about and a fulfillment of the Progressive Era?

The New Deal wasn't new: it simply used many ideas that had become popular during the Progressive era some thirty or forty years earlier. However, it really was new in that it implemented these reforms in a large-scale way, at a national level. Prior to the New Deal, these ideas mostly had only been talked about.
The New Deal was a monumental shift in democratic government because, for the first time, the federal government was understood to have as one of its core functions ensuring social welfare: seeing to it that Americans had the minimum food, housing, employment, and legal protections that helped make possible a reasonably decent human life for most people. It fulfilled such Progressive dreams as a forty-hour work week, a minimum wage, an end to child labor, and Social Security. By the late 1940s, once World War II was over, progressives such as Dorothy Day, who began the Catholic Worker movement, could stand back with satisfaction and see that much that they had marched for, been jailed for, and campaigned for, such as the forty-hour work week, had become a reality and that average Americans were prospering in a way that they never had before.
In contrast to FDR, who believed the federal government needed to step in and do everything possible to alleviate the suffering brought on by the Great Depression, Hoover believed government should be kept small and that markets—private industry—and private charity should work out the problems brought on by Depression. He was not against government interventions and did provide federal money for public works projects such as the Hoover dam, which employed people. He also started the Reconstruction Finance Corporation to keep important industries from going bankrupt. However, he also believed that tax cuts would stimulate the economy and believed the country would "bounce back" on its own. He gave heavily to private charity and encouraged other Americans to do so to help the hungry and homeless; however, this patchwork charity was inadequate to the crisis at hand, which required a fundamental rethinking of the role of government in civic life. In a nutshell, it could be said that FDR grasped the magnitude of the crisis brought on by the Great Depression in a way Hoover did not.