When BaTiO3 is cooled below the Curie point (120°C), the cubic structure changes slightly, so that $Ti^{4+}$ shifts by 0.006nm away from its position at the center of the cube. I don't know why this happens, but it makes the material piezoelectric.
I suppose that CaTiO3 has a very similar structure, but it is not piezoelectric. Does anybody know why this is so?