Probably the most confusing aspect of these coordinate systems is the difference between math and physics with respect to the polar angle and the azimuthal angle. The polar angle is the angle measured down from the z axis. The azimuthal angle is measured around the z axis.
You can tell which system an author is using by which angle shows up in the z-coordinate. The z-coordinate, in spherical coordinates, always contains the cosine of the polar angle, and is independent of the azimuthal angle.
Does this clear some things up?