Karp-like reductions vs Cook-like reductions for Functional Complexity Classes. Assume we have two counting (functional) complexity classes $A$ and $B$. Suppose that


Under Karp-like reductions $A$ is strictly inside $B$.   
Under Cook-like reductions $P^A=P^B$.



  What does this tell us about the relationship between these two
  counting complexity classes?
I think this situation is too general to draw many conclusions, but here goes...

If $A$ is closed under Cook-like reductions, then (2) would imply $B \subseteq A$, contradicting (1), so it tells you that $A$ is not closed under Cook-like reductions.

One way to paraphrase the original statement is: to make the classes equal in power requires more than a single query. 

One can then ask about other intermediate types of reductions to get a better sense of just how many queries are needed and in what way e.g. are the classes equivalent under truth-table (nonadaptive) reductions? What's the best bound we can put on the number of queries ($2$, $O(1)$, $O(\log n)$?) Since these are counting classes, one could also ask about parsimonious reductions.