Sulphur is the element. The atom.
Sulphates are compounds that contain the SO4 ion. Sulphur bound to four oxygens.
Sulphites are compounds that contain the SO3 ion. Sulphur bound to three oxygens.
There's also Sulphur Dioxide, a molecule which seems to have something in common with the sulphites in various ways.
Various sulphites and sulphur dioxide seem to be inimical to life, and this makes them good as preservatives. If you put them in food, the food will not go off so fast, because it's poisonous to various bacteria that would otherwise eat it.
They're supposed to be harmless to humans, because we have an enzyme, sulphite oxidase, that turns them into sulphates, almost immediately they're eaten.
That's why they're permitted food additives. They also occur naturally in fermented products such as beer and wine and sauerkraut. Normal humans can eat all these things without the slightest trouble.
There seem to be a lot of people who don't react well to sulphites, including me.
These people get terrible hangovers from wine and beer. Headache and nausea. They get similar hangovers from sauerkraut, and lime cordial, and preserved fruits. They can usually drink spirits without too much trouble.
I think we may be short of sulphite oxidase for some reason, making sulphites a poison.
Medical "science" tells us that this problem does not exist. It does accept the existence of sulphite allergy, but that's supposed to be very rare. Read into that what you will.
Sulphur is essential for life, you can't do without it. But some people claim that high-sulphur diets make them ill. Read into that what you will.
Eggs contain sulphur, in various forms. Probably not sulphites though. I eat lots of them with no effect.
I think that the only advice I have to give is: “If eggs make you ill, avoid eggs. And try to find out why.”