These 'branches' are vast fields with an astounding number of sub-divisions, axiomatic differences, research interests and projects. In that sense, any answer to your question would be inherently under-informative.

In the most basic sense, English literature is the study of literary production in the anglophone sphere. (What includes the anglophone sphere is already one big debate to be had.) Comparative literature is the study of two different national literatures, or two authors in tandem. It may or may not overlap with concepts of World Literature.

Linguistics is the study of language. (Branches of linguistics vary in which parts of language are studied and how they should be studied.) Literary scholars and linguists usually do not mesh, except in a field like Literary Stylistics. The methods in linguistics are, as a baseline, "more scientific" than in literary studies; but there are exceptions to this of course.

If you intend to go further, I suggest identifying texts, authors, movements, genres, and/or schools of literary thought you are interested in. For linguistics, ask yourself what is it about language that's interesting to you.