Does the black body emit more than any other type of body?

I found this on Wikipedia article on black bodies:

A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition.

Now I'm a bit confused. Isn't the black body radiation given by $\frac{P}{A}=\sigma T^4$

So it doesn't depend on body color, composition or anything else than its size. Shouldn't the radiation be the same for all bodies, but we use the black body because its a convenient model because all of its radiation comes from emission and not reflection or anything else?