Do relativized relations between complexity classes tell us anything about the nonrelativized relation?
I am by no means an expert, but 2 things come to mind:

Because if we know things like $A^L \neq B^L$, for some $L$, then a proof of $A = B$ need to be non-relativizing. This puts constrains on the techniques used for such a proof, and thus it is helpful in the way towards getting a proof.

Because if you prove $A^L = B^L$, and the oracle is not very strong, then it seems that you are narrowing down the possibility of $A \neq B$, as with some extra help they become equal. This can arguably be evidence towards believing that $A = B$. To me it feels like saying that John and Peter were equally strong in a fight with boxing gloves. That doesn't necesarily mean that they will be equally strong without the gloves, as one of them could be weaker but use the gloves better. Nevertheless, the information about their fight with gloves is arguable evidence of their relative strength.