Rob,
This is a long story. I'll try to do a short version of it.
At every, considered to be, sacred Christian site in Israel there are generally 2 churches. The Roman Catholic and the Greek Orthodox church. In this spot there are 3 with the Armenian Apostolic Christians. All share the same building.
News media like to make it a religious war but it is more of a ethnic or people group war. You will notice that they do not say "Roman" or "Greek." Why? Because that gives away what it is about. Roman Catholics are a mixture group of people but I understand that it is mostly Palestinian. The Greek Orthodox are of course Greeks.
They fight for control of the site all of the time. Has been that way since the beginning. I've seen them spit at each other. I've seen a funeral procession have to wait outside for over an hour because there was, from what I understand, an impromptu service with maybe 10 people in attendance. They continually do things to harass each other.
But don't ever think it is an issue of religion. They beliefs are almost exactly the same. They just don't follow them very well.