I think they are not the same thing, not exactly, and the difference is interesting.
Hydrogen peroxide is a rather - comparatively - unstable substance, it decomposes to water and oxygen; hydrogen peroxide is a rather strong oxidizer, which defines a lot of its properties.
I'd assume double -O- bond in peracetic acid behaves the same - decomposes with release of O and acetic acid (or the anion of the acid). This oxidizing effect likely provides the effect which is desired - the same which chlorine would produce, that is, oxidizing a lot of things in chickens making them safer.
I've heard that hydrogen peroxide is used in Europe instead of chlorine in US for water treatment - for example, in swimming pools. I'm not sure why peracetic acid is chosen.
They are not chemically the same thing. Presumably they don't use vinegar because vinegar is less effective. However, the implication is that the chemical they use will break down to vinegar by the time it gets to the consumer.
Why aren't they just using vinegar if it's the same thing?