The 1960s are more commonly compared with the 1920s, but there are some comparisons to the 1930s that can be drawn.
The major similarity is that both of these decades were times of increasing government involvement in the economy. In both decades, the Democratic Party was fairly strong. In the 1930s there was, of course, the New Deal. These programs inserted the government into economic life in unprecedented ways. In the 1960s, President Johnson declared the “War on Poverty” and his desire to build a “Great Society.” These initiatives built on the New Deal and got government even more involved in the economy.
The similarities largely end there, however. The New Deal came about because of desperation. By contrast, Johnson’s initiatives came about because of widespread prosperity. This is a major difference.
The 1930s were a time when the United States was isolationist. It had been through World War I and did not want to get involved in another war. Therefore, the US did not do much internationally in this decade. By contrast, the 1960s was a time of intense US involvement in foreign affairs. This was the decade of the Bay of Pigs, the Cuban Missile Crisis, and the Vietnam War.
The 1960s were a time of major social ferment. The Civil Rights Movement and the counterculture/anti-war movement made this a very unsettled time socially. Of course, the 1930s were unsettled, but they were unsettled in a completely different way. The 1930s was a time when people were having to adapt to the Great Depression. They had social change forced on them whereas the social change of the 1960s came about voluntarily.
Thus, these decades have a little in common, but not a great deal.