Amongst different foundations of mathematics, $ZF$ and $NF$ are talking about "sets" but $MK$ and $GB$ are talking about two sorts of objects "sets" and "classes".

What are benefits of studying the axiomatic systems with more than one sort of objects even if they are conservative relative to $ZF$? Is it just because theorems about proper classes have simpler forms in such systems in comparison with corresponding theorem schemas in $ZF$ or it is because we lose some useful information and power of the theorems on proper classes when we reduce them to $ZF$? If the last is true, what are the examples of such harmful reductions? Does the same phenomenon happen for sets as same as proper classes too? In the other words, do $GB$ or $MK$ in some sense tell us "more" about sets and proper classes than $ZF$?