The main difference between the Communicative competence and the Linguistic is that the linguistic competence belongs to the whole Communicative Competence or Language Knowledge that the students or native learners have.So, Linguistic is a component from Communicative competence in other words.
The whole issue of communicative context is a very interesting one both as a reaction to Chomsky's discussions of linguistic competence and performance, as a primary aspect of communicative competence, or as an extended subject in its own right in Ethnography of communication, Goffman's frame analysis, or other aspects of context in light of meaning-making and interpretation. However, one direct answer to your question above is that communicative context is not really related to linguistic competence if you are referring to Chomsky's use of the term/concept of competence. Chomsky had a different agenda -- that of demonstrating the generativeness of language -- specifically syntactic/grammatical facets of language and in doing so created a model of the idealized speaker/hearer to discuss his abstract concepts....his was actually quite divorced from the actual considerations and use of language by actual individuals. I think that there are so many authors (including Chomsky) who have discussed this. I think Fred Newmeyer's editions (especially the 2nd edition) of "Linguistic Theory in America" has some excellent discussions of this. Chomsky focused on the underlying abstract rules of grammar and the characteristic of generating acceptable grammatical constructions while NOT generating un-acceptable ones...he focused on the idealized speaker/hearer because he was working in theory not application and divorced his conceptions from their use in interaction.....it is one of the reasons Dell Hymes first published and discussed "on Communicative Competence"....back in ...68 or 72...So, from a Chomskyean perspective and use of the term, linguistic competence has virtually no relation to communicative context.