I've just been reading up on this now, and it seems that flexoelectricity is distinct from piezoelectricity, but they are related. Flexoelectricity involves the strain a material undergoes — i.e. how much the material deforms — whereas piezoelectricity is about the stress a material experiences. Of course, stress and strain are pretty closely linked — applying stress to a material often induces strain.

I need to read more about this, but I wonder whether there may be some materials that are piezoelectric but not flexoelectric, and some that are flexoelectric but not piezoelectric, as well as some materials that are both.