Both are types of porcelain. Historically, animal bone ash was mixed into clay - giving it the name 'bone china' - to make porcelain. However, today's technology allows us to forego this bizarre, outdated (yet still widely used) method.
Poppy Angeloff teacups are made out of 'fine china': a cruelty-free modern porcelain that is just as beautiful and sturdy, but way friendlier to our furry friends.