Well my question is quite simple and straightforward. What is the key differences between Private Beta, Public Beta and Full Site? Are they only different by the statistics shown in Ara51? Are newer functionalities added to when the states change?
The private beta was temporarily available only to those that signed up on the proposal on Area 51. After that the site goes into public beta, meaning that everyone can join.
Public beta is the staging area where it's found if the site is feasible enough to make it to a full site. On a beta site the reputation requirements for different privileges is quite a bit lower than if it were a full site.
As for functionality, I can't recall there being any major differences between beta and promoted site.