Given that I've no allergies or sensitivities to egg or whey based proteins, I've not noticed any measurable differences in the area of gas. However, it does seem to have more to do with the quantity of protein ingested at one time than the type. Regardless of which you use, if you don't chug the entire glass at once, you reduce the incidence of toxic fumes.
As to other differences, egg (white) protein is more balanced in the amino acids it provides as well as being more effectively and quickly absorbed into your system than whey protein. I can't remember the advantage that whey has (though I know it does have some.)
Personally, I mix both proteins so that one gets absorbed fast and the other more slowly. I just make the shake last an hour rather than 2 minutes and that seems to have a positive effect on the atmospheric environment around me.