We know the word Peritonitis from emergency removal of an appendix. This is a very serious infection that invades the Peritoneum which is the lining of the intestines. In the amount of time it would take a surgeon to say, "The appendix burst so I took it out", Peritonitis has already set in and the infection has started to spread. It travels so fast through the intestinal lining, that it hits the blood stream in no time and then that's sepsis, no easy illness to clear up. (Incidentally, one of the major antibiotics used to cure peritonitis is Flagyl (Metronadazole).
What does this have to do with egg peritonitis? Nothing at all. It's a totally different thing.
We have to start with the ovary and the oviduct. A yolk is formed in there first. The shell is formed afterwards to enclose the yolk. If the bird is deficient in calcium, one of three things can happen:
1) The shell forms but is thin or only partially forms.
2) The shell forms and is thin and cracks inside the body
3) The shell never forms.
Calcium is very, very vital for the hen.
If the shell is weak and cracks inside or it doesn't form at all or it forms only partially, the oviduct is stuck with a yolk and it's going to flow out into the body cavity. It will pour over the liver, the kidneys and into the abdominal cavity. It's going to kill organs because this stuff turns toxic. This is called egg peritonitis. It has nothing to do with the intestinal lining and it is not egg binding.
Egg binding happens when the hen can't pass the egg. It can be because of a weak or cracked shell or some other obstruction in the oviduct like a tumor or another egg that failed to pass out. An egg bound hen doesn't necessarily have to have a yolk floating around in the body somewhere so egg peritonitis doesn't have to be there at the same time the egg is bound or there can be both at the same time.
How can you tell the difference between egg peritonitis and egg binding. The layman generally can't. You might be able to feel an egg shell inside but that isn't going to tell you if the yolk is free in the body. If you see a piece of yolk passing in the dropping, you could guess that there is a yolk loose in there but you don't know if there is peritonitis or if the yolk is being fully ejected through the vent. It takes a very skilled vet to be able to distinguish between egg binding and egg peritonitis or see both on an X-Ray and sonogram.
Eggs that don't pass can either calcify more or can be reabsorbed by the system. More eggs can form and back up, start to fill the body cavity, pushing into vital organs and causing their death.
Either way, egg binding and egg peritonitis are both killers and are equally serious problems that require a very qualified avian vet to make the examination yesterday, not today because today could be too late.
Recommended Further Reading: http://www.birds-online.de/gesundheit/gessonstiges/legenot_en.htm