The main way in which the federal government changed during the New Deal was by expanding greatly and expanding its role in American society.
Before the New Deal, the federal government was a relatively small entity. It was not expected to do very much and it did not have a huge number of employees. People were generally expected to fend for themselves. The economy was supposed to be left alone to fix itself in bad times. This was a very limited government.
During the New Deal, this all changed. The government inserted itself into many areas of economic life that had never been contemplated before. It involved itself in providing jobs for people who did not have them. It involved itself in trying to ensure that retired people would have enough money to live on. It involved itself in trying to ensure that people would not lose their money when banks failed. It tried to completely remake the physical environments through programs like the Tennessee Valley Authority. These are all things that had not previously been expected of the federal government.
President Franklin D. Roosevelt's New Deal programs substantially increased the size of the federal government while simultaneously greatly expanding the government's role in the daily lives of much of the populace.
While the Constitution of the United States established the structures, roles, and responsibilities of the federal government, it allowed the possibility of the government's expansion in the years following ratification. With expansion came the increase in the federal government's powers—a development possibly anathema to the ideas of limited government, which, ultimately, the New Deal and future president Lyndon Johnson's 1960s-era "Great Society" essentially supplanted.
Prior to the onset of the Great Depression, the catalyst for the New Deal programs that were created during the 1930s, the size of the federal government was fairly limited. The catastrophic consequences of the depression, however, required measures that were unique to the situation. However, these measures fundamentally changed the government's role relative to that of the population it purportedly served. Roosevelt understood that the extremely high level of unemployment coupled with the collapse of the banking industry necessitated extreme measures. Those measures included the establishment of a number of agencies and departments designed to put people back to work while improving the country's infrastructure and industries. New Deal agencies like the Civil Works Administration, the Civil Conservation Corps, the Public Works Administration, the Tennessee Valley Authority, and others were all intended to provide jobs while constructing infrastructure like roads and dams. That some of these programs continue to exist today is testimony to the resiliency of federal bureaucracies.
The federal government is much larger today than it was prior to the New Deal. It plays a much more prominent role in the national economy and the programs it created remain living testaments to the will of Roosevelt and his cabinet in expanding the government's presence in the lives of the public.