There is very little that was common to these governments during the 1920s.
The only point of comparison that can be made very well is that both could be at least somewhat repressive. The Soviet Union, of course, was quite intolerant of any dissent. At least in the early 1920s, the US government, while much more tolerant than the Soviet Union, was in the midst of a “red scare.” This led to some degree of political intolerance of radical ideas.
However, this similarity was slight and the differences were major. The Soviet Union was not democratic; the US was. The Soviet Union tried to manage its entire economy through government plans. By contrast, this was the age of laissez-faire in the United States. These were fundamentally different systems of government.
The federal government of the United States during the 1920s was notable for a conscious turn from the Progressive politics of the first part of the twentieth century. The presidents of the twenties, particularly Warren Harding and Calvin Coolidge, took a pro-business, anti-labor stance that was demonstrated by Coolidge's famous statement that "the business of America is business." So the federal government took a laissez-faire stance toward governing, cutting income taxes and relaxing regulations. However, much of this was rhetorical, and neither the Republican presidents nor the Congress did much to eliminate many of the institutions put in place by the Progressives.
If the 1920s saw at least a theoretical decline in government activism in the United States, the situation was the opposite in the Soviet Union. Lenin died in 1924, and the struggle to replace him accompanied a significant increase in the power or the Soviet state. As Joseph Stalin emerged as leader of the new revolutionary state, a process which was complete by late 1920s, he began to implement many reforms aimed at modernizing the state. While these reforms would not really take root until the 1930s, they began in earnest in 1928. So the 1920s in the Soviet Union saw an expansion and consolidation of the powers of the Communist state.