Well, my opinion is, that those processes were more different than similar. Still, I think it is possible to find similarities even between apples and oranges, so, let me try.
It was in years 1933 to 1936 when President Franklin D. Roosevelt passed a group of domestic programs. They were all about stimulating the devastating economy of a country after The Great Depression (1929-1939). The basis of those programs was down to three main ideas, namely relief, recovery, and reform. The Progressive Movement came first, and was before The New Deal, in time from late 19th century to early 20th century. It is important to remember; it was a time when political, cultural and economic areas were rapidly changing due to the Industrial Revolution.
Both programs, The Progressive Movement and The New Deal, were improving the quality of people’s lives, so they were actually earning money while developing the country. Comparing to the time of The Great Depression, when people had no food or jobs, it was a great breakthrough.
Also, to be clear, I have to mention the differences. The New Deal was a lot like the start of the global recovery. It helped people to get food and jobs, so they were actually building the country by working on bridges, roads and railway tracks. The Progressive Movement was more of a continuation of the improvement and had pushed the wealthy class further beyond the lower class. By that time people usually had jobs and food, so the country was already moving full stream. The New Deal, of course, had a lot of progressive roots, but the goals were completely different. While by the time of The New Deal it was all about surviving, The Progressive Movement has brought possibilities to live or to live better, if said shortly. The first program was a needed response to crashing and poor time, but the second one was already a conscious movement toward a booming economy.
The similarity of those programs was in fact, that government was involved in business, and there was a lot of industry regulation during both periods.