I have always said that both of these doctrines are very similar. For starters, they are very controlling and are very dogmatic.
(1) Followers of both doctrines like to control people. In North Korea, for example, hidden cameras are used to spy on people in the streets. A man with long hair could easily get arrested.
(2) Censorship. They both censor whatever offends them. Let's not forget that during the Cold War, most forms of music were banned in the Iron Curtain, and let's not forget Christian organizations were constantly attacking different types of music
(3) Extreme altruism: communism took sharing to an extreme, by forcing farmers to share their belongings with other farmers to create equality. Communism's tendecy to abolish social classes was no different than Jesus preaching against the rich. Or one of the ten commandments saying you should never covet thy neighbor's goods.
(4) Persecution of gays, feminists, etc.
(5) A tendency to turn followers of their dogmas into saints. Typical of the Catholic Church, but the communists did canonize Stalin and Lenin, as if they were saints.
(6) Genocide. Christians like to say communists killed people because they were atheists, but the truth is, communists destroyed anything or anyone that went against their doctrine. The conquistadors, Charlemagne, Oliver Cromwell were no different.
(7) Control of the educational system. Communists tried their best to convince their young that their way was the only way. Schools were used for this. Christians do the same with the educational system.