The "three wavelength correction" information that edz supplies is a correct description of traditional APO design. It doesn't guarantee, however, that the term APO is used consistently by manufacturers to mean exactly that, and the degree of difference between APO and achromatic designs varies as well.
My only experience with a direct comparison of APO vs non-APO within a brand was with 180mm focal length lenses for Leica R, comparing their 180mm f:3.4 APO aerial reconnaissance lens (designed for military, but made available to the general public) to a regular 180mm lens. I don't recall if the regular lens tested was their f:4 or f:2.8, it was 30 years ago. The comparison shot was of the back of a car across a parking lot shot on Kodachrome 25 and the camera on a tripod in full sun, with specular chrome highlights off the bumper. There was a sticker on the bumper with fine print, a parking permit with multiple colors and black text. That sticker was much better defined in the APO shot, with no color fringing and text that was much easier to read. The non-APO shot wasn't bad, but at high magnification, the APO clearly outperformed it.
That particular 180 APO was designed for performance at greater distances (being an aerial recon lens) and Leica provided their 180 f:4 with a helical that focused closer because it performed better at short distances.
An APO should perform better than a non-APO from the same maker in both B&W and in color, but not all APOs are designs that truly fit the classic definition, being more ad hype than achieved design goals. APOs often contain extra corrective elements, special refractive index glass, or aspherical elements to achieve APO performance. That drives the prices higher. There are also other aberrations besides chromatic that contribute to a lens' performance, and those have to remain well corrected.
The term APO has been stretched (if not abused) in many products, but many others do meet the classic definition, and are clearly better than their achromatic counterparts. The proof is in the negative/transparency. Sorry I can't help you with direct experience or second hand reports on the Mamiya gear.
Lee