It absolutely will change the "breathing" characteristics of the mattress. How dramatic that change is depends on how breathable the mattress is and how breathable the pad is.

But, yeah, as /u/Rhynchelma said there's all sorts of ways mattresses get wet, from their list to eating in bed, sex, pets, could go on and on. And it's not like a mattress can be put in the laundry.

That said, many mattress covers are garbage. It's a tough product to make well. Performance rain pants last a much shorter length of time than performance rain jackets because of all the extra flexing and rubbing the membrane is subject to. Mattress pads get that flexing and wear 8 hours a day, every day.