As far as I can understand, $E_\infty$ ring spaces are homotopy commutative H-spaces. If we have a space, and we have an action of an $E_\infty$-operad on it, then we say that it has a homotopy commutative multiplication?
This seems to be what people say, however, there must be more right? In the wikipedia article on $E_\infty$-operads, it is mentioned that what "$E_\infty$" gives us is a multiplication that is associative and commutative up to "all higher homotopies." It also says that an $E_2$ space is a homotopy commutative $A_\infty$ space, i.e. if all we want is a commutative multiplication, up to homotopy, then we're done after the second level. What is the significance of all these higher homotopies? I used to think the point was that we just wanted a way to get homotopy commutative spaces.