Currently, the only distinction between alliance government types is whether they allow their members to leave the alliance freely and whether the members are able to start a revolution.
When it comes to leaving the alliance, the government types 'democratic, 'socialist' and 'monarchy' give the greatest freedom to the members. In such alliances the members may freely leave the alliance without leader's approval after a 72-hour waiting time (24h for Mossopps). This is different from the government types 'dictatorship' and 'communism' that force members to plead for a permission to leave the alliance. A member is unable to leave such alliance if the leader does not either give a permission to leave or kick the member.
The other distinction is the ability to start revolutions. A revolution is not possible in an alliance whose government type is either democratic or socialist. The other three types of government (monarchy, dictatorship, communism) give the members a possibility to start a revolution which could thrust the alliance into state of Chaos which dramatically cuts down functionality of the alliance for a period of time and can ultimately lead to turning the alliance into a democracy.