NL and NP compute different binary relations, so what?
We know that $NL \subseteq NP$, but that $NL \subsetneq NP$ is not known. So one cannot outright say that $NL$ and $NP$ are different. 

We only know for sure $NL \subsetneq PSPACE \subsetneq EXPSPACE$ by space hierarchy theorem. And $P \subsetneq EXPTIME$,and $NP \subseteq NEXPTIME$ by time hierarchy theorem. 

We only know for sure is 
$$L\subseteq NL \subseteq P \subseteq NP \subseteq PH \subseteq PSPACE =NPSAPCE \subseteq EXPTIME \subseteq NEXPTIME\subseteq EXPSPACE$$ 

Only above four inclusions are known to be strict. All the other inclusions are unknown and open problems. (Though most but not all have unproven belief that each of them is strict).

See https://en.wikipedia.org/wiki/PSPACE#Relation_among_other_classes.

Also, I think, we might be able to prove that there exists a language $B$ for which $NL^B \neq NP^B$. Probably $U_B$ and $B$ of Baker, Gill and Soloway will work here too. And we are also able to show that $NL^A = NP^A$ for $A = TQBF$, the $PSPACE$ complete problem to decide the true quantified Boolean formulas.