It is known that ZFC needs infinitely many axioms, but NBG (Neuman-Bernays-Gödel set theory) is finitely axiomatizable (as first-order theories of course). But both theories agree completely on the set part of their universe (as far as I have read).
How could this be? How can describing even more objects (proper classes in NBG) while keeping the complexity of some part can reduce the effort to describe this structure? Is there some plausible and evident explanation of this observation? Maybe some philosophical insight from someone who knows the proofs of these statements?
Maybe, is it because the proper classes allow NBG to quantify over predicates in some sense? Something for which ZFC usually needs axiom schemas? If so, why isn't NBG absolutely favorable to ZFC as foundation of math? I mean we also prefer set theory to Peano arithmetic because the latter one allows us to quantify over subsets (in some sense) despite it is a first-order theory (I know we prefer ZFC over PA for tons of other reasons too).
I know of this and this question, but I ask specifically why the finite axiom system is not a convincing reason for NBG. However,the question on which to prefer, ZFC or NBG is secondary. Please concentrate on the finitely vs. infinitely many axioms part and how this can be.