In both the 1930s and the 1960s, the role of the federal government in bettering the lives of average citizens increased.

In the 1930s, the role of the government shifted fundamentally. Before the New Deal, social welfare was considered the province of the states. The role of the federal government was primarily focused on warfare and defense issues. The New Deal, for the first time ever, established a host of programs and laws that, at a federal level, improved the lives of average citizens, such as federally secured bank deposits, the forty-hour work week, and Social Security.

Johnson's Great Society programs did not, unlike the New Deal, fundamentally shift the relationship between the federal government and the states. They also did not challenge a national consensus in which a robust social welfare state existed along side a robust capitalist economic system. The programs that emerged in the 1960s primarily built on and expanded the platform the New Deal had already established. For example, Medicare was a major add-on to Social Security that was implemented under the Johnson administration, but the Roosevelt administration had already established the precedent that the federal government should have a role in insuring the well being of the elderly.

One enormous difference between the social programs unrolled in the 1930s and those of the 1960s was context. The programs of the 1930s were a response to a desperate economic situation in which the economy had all but collapsed by the time FDR came to power. If the government hadn't pumped money into the economy, there is no saying what might have happened. Banks were failing, the stock market was all but dead, unemployment was extremely high, many average people had lost everything, and many more were hungry, homeless, and hopeless. FDR is largely credited with staving off either a communist or a fascist takeover.

In the 1960s, the United States was at the height of prosperity. The standard of living had risen dramatically during the 1950s, and many were living at level that had once been beyond their wildest dreams. The government increased its spending on social programs in part because the tax dollars piling up in the treasury were a drag on economic growth.

Another key difference between the 1930s and 1960s was the intense focus in the 1960s on marginalized and oppressed groups: blacks, Native American, Latinos, gays, and women began clamoring for equal rights in ways they did not in the 1930s. FDR had sidestepped black rights so that he could hold together a winning coalition that included the southern states. Johnson, in contrast, went full throttle into supporting civil rights. Johnson worked more fully and intentionally to insure minorities had the same opportunities as whites.

During both periods the Democratic Party was quite strong.  There were also social tensions—in the 1930s, these were caused by the Great Depression, and in the 1960s, they were caused by the civil rights marches.  Domestically, the 1930s saw the rise of the New Deal, which was an attempt to alleviate the twenty-five percent unemployment rate Americans faced during the Great Depression.  In the 1960s, Lyndon Johnson, a follower of Franklin Roosevelt, rolled out his own Great Society programs which were supposed to help the poor through programs like subsidized housing and Medicaid.  Roosevelt ran into a Supreme Court which questioned the constitutionality of his programs, while Johnson's anti-poverty efforts were ended largely by the Vietnam War.  

The United States did undergo protests during the 1930s concerning the rampant poverty in the nation—one of the more famous protests was carried out by the Bonus Marchers.  The government's response to the veterans' requests for help assisted in ending Herbert Hoover's political career.  The Democratic Party was slow to help the civil rights marchers in the 1960s for fear of losing their Southern political base, but after the protests were televised, they soon reversed course by passing the Civil Rights Act of 1964.  Other than the consistent fact of decades of Democratic Party dominance at the federal level, there are not many good comparisons here.  

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.