Specifically I'm interested in the split between Structural Case and Morphological case.
Structural Case has been part of Chomskyan syntactic theory since at least Government & Binding (GB). Roughly, Case Theory in GB and the minimalist program state that all nominals in a grammatical sentence must have Structural Case and they move away from their base positions to get it.
Morphological case, or m-case, is the more familiar case (nominative, accusative, dative) that surfaces phonologically on nominals in languages which do so. The most recent theories of m-case that I am aware of derive from Alec Marantz's 1991 paper "Case and Licensing" (which was later reissued as an intro to Reuland (2000)).
When I last checked, Nominative and Accusative were assumed to be Structural Case and the rest were sort of up in the air.