Maoism has various elements not found in Russian and European Marxism-Leninism. In particular, while stressing that an agrarian society has major disadvantages in competing with an industrialized society, it still emphasizes agrarian society as a major factor in a communist revolution.
Furthermore, while Maoism takes pride in aggressive opposition to traditional, Chinese, Confucian values, my impression is that it has often carried the idea that the group is above the individual to extremes that are not even found in Soviet Russia, and in that way appears to be more culturally traditional than it likes to admit.
Marxism-Leninism is a subset of Marxism
Marx was not a “Marxist”, he was a philosopher and social scientist. His practical political activities occurred at a time when organization of the workers was unheard of, and before the various schisms between socialists, social democrats, communists, and so on. They were revolutionary in the 19th century, but from today’s point of view, are nothing more and nothing less then democratic.
The Bolsheviks were not interested in philosophy, they were interested in revolution and seizing power. They took from Marx’ ideas what they could use to that end, bundled them together with Lenin’s writings (which were much more practically oriented to his particular historical situation) and calcified them into the ideology of a highly bureaucratic state, condemning themselves to intellectual stagnation, very much unlike Marx.
Marxism is a subset of communism
Communism is a mere ideal of a society where human beings can be free and equal, by abolishing private ownership of the means of productions (factories, natural resources, infrastructure), thereby abolishing the ruling classes, and replacing them with a society in which everybody has the same material means and an equal say in the political and economical decisions of that society. To my knowledge, no state has ever claimed to have actually achieved this.
Communism is a subset of socialism
Socialism is a very broad term, traditionally applied to a society where significant industries are state controlled. The political right finds pleasure in using it as a fighting word, applying it to any country where not every last nursing home, fire station and toll road is operated for private profit. For communism, see above.