The difference in the groups, is which animals you will get back. You will get back what each person signs up for.
Punch art is made with punches. Paper piecing can involve many elements to compose the piece. We are using punches to make our animals. Some may call it paper piecing as well, but I want to ensure that we are following the Punch style, so that anyone can replicate... you just need the punches... no drawing etc.