Each commands the loyalty of more than a billion people. Each is governed by a leader chosen in secret by a small group of its senior members. Each lays out a set of moral guidelines that its followers should live by. The Vatican and the Chinese Communist Party have quite a few things in common (apart, of course, from a belief in God...)
But the two sides have been in conflict for decades. Diplomatic relations were broken off after the Communist Party came to power in 1949 (instead of recognising Beijing, the Vatican is currently one of 23 states that has diplomatic ties with the government in Taiwan - the island that China considers a breakaway province.)
Right now, the China-Vatican argument comes down to this: who gets to pick the church's bishops in China. The Communist Party says it should be China. The Vatican says it should be the Pope. It's an important fight - because there are around 12 million Catholics in China. The argument over control has split the Catholic Church in China into two - there's the official state-run church (bishops picked by Beijing) and the unofficial - or underground - or church whose parishioners are loyal to the Pope.
But things may be changing. Some ultra Sino-Vaticanology for you...
1) In June 2007 the Pope sent a 55-page open letter to Catholics in China - a kind of (lengthy) goodwill greetings card. Benedict XVI stuck to his bottom line - the Church in Rome gets to pick all bishops anywhere in the world. But he reached out to all Catholics in China and suggested that the Vatican was open to negotiations with the Chinese government.
2) In September 2007, a new Bishop of Beijing, Joseph Li Shan, was consecrated. The bishop was chosen by the Chinese government - but Vatican watchers say that the Church in Rome quietly approved of his appointment.
3) And just a few days ago (8 May) the China Philharmonic Orchestra played a concert for Pope Benedict XVI in the Vatican (it performed Mozart's "Requiem" and a selection of Chinese folk tunes.) China says the visit was just about music. But you clearly don't allow your national orchestra to play for a leader whose state you don't have ties with unless you've thought it all through beforehand.
At the moment, the Chinese media in Hong Kong (which has more freedom to report, rumour, and guess than the media in mainland China) speculates that all of the above means that the Vatican and Beijing may be about to re-establish diplomatic ties.
But one veteran Rome-watcher has just reminded me that the wheels inside the Vatican move incredibly slowly.
As they do in China as well.