Summary: We consider computing the difference between two Horn theories. This problem may arise, for example, if we take care of a theory change in a knowledge base. In general, the difference of Horn theories is not Horn. Therefore, we consider Horn approximations of the difference in terms of Horn cores (i.e., weakest Horn theories included in the difference) and the Horn envelope (i.e., the strongest Horn theory containing the difference), which have been proposed and analyzed extensively in the literature. We study the problem under the familiar representation of Horn theories by Horn CNFs, as well as under the recently proposed model-based representation in terms of the characteristic models. For all problems and representations, polynomial time algorithms or proofs of intractability for the propositional case are provided; thus, our work gives a complete picture of the tractability intractability frontier in the propositional Horn theories.