User blog comment:LittlePeng9/First order oodle theory - clarification/@comment-172.56.15.104-20161020231854/@comment-1605058-20161021155741

Well, to be honest, I see more differences between Rayo's and Busy Beaver functions than similarities, so I am unsure why such a question comes up.

The first and most significant difference is that they are two very different functions defined in very different ways. Busy Beaver function employs the notion of a Turing machine and counts the maximal running time of such a machine with bounded number of machine states. Rayo's function, on the other hand, uses the formal language called "first-order set theory", in which we can express statements about the universe of all sets, and then the function returns the largest number which can be defined using this language. I hope this answers both questions.