I think the origins of the divide have much to do with language and the development of the University, which in turn relate to much broader issues. The predominance of English in so-called analytic philosophy is not happenstance; that analytic/"anglo-american" philosophy grows out of and thrives in the research University isn't either.
I also think the conception of the divide as being between two traditions or schools is highly questionable. An analogy of twentieth century "continental" philosophy and twentieth century music might illustrate this. It is often assumed that what the so-called second Viennese school began (especially Anton Webern) was taken up and brought to further development by the post-war serialists (e.g., Boulez, Stockhausen, Ligeti, et al.). In a similar way, continental philosophy is supposedly a lineage of German idealism, phenomenology, etc. taken up and worked out by, again, post-war (primarily French) advocates. The issue in both cases is whether, to what extent, and in what sense, the thought and work of these earlier thinkers and composers is actually engaged or "developed" by those who claim the inheritance after the war. I would argue that they leave untouched (and possibly distort) far more than they have received or taken up.
So, I don't think there is much of a rivalry between analytic and continental, and what remains has faded, primarily because there are no longer two combatants, if there ever really were. The current status of philosophy departments and what and how they research has far more to do with fundamental features of the modern world than any petty rivalry between two academic camps.
