If you ground a structural color material down into a dust, it would no longer produce the same color effects anymore. By contrast, pigments and dyes have their absorption at the molecular level and would only lose their effect if you modified their chemical bonds.
With structural color, the substances themselves do not have this spectral absorption characteristic. Instead, very specific configurations are needed to produce different effects. Imagine microscopic arrays of prisms and mirrors.
Also, structural colorization is often very angle-dependent. The iridescence of flowers, feathers, etc. comes from this characteristic. At different angles of incidence of light reflecting to an observer, different colors become visible. Pigments do not do this.
Also, I think there may be differences in usage of these terms between physicists, chemists, and biologists. While writing this, I also realize I am not sure if the terms become ambiguous for relativelt inert, microscopic crystals. From a physics standpoint, I think the crystals may exhibit structural color and pigment effects. From a practical standpoint, they might behave a little more like a pigment in that you could have what seems a fine dust and mix it into paint. But, if you dissolved it to separate the crystals back into free molecules, it would then only exhibit the pigment effects.