Stopping power is generally used for charged particle interactions, whereas the mass energy absorption coefficient is specific for photons.  If you look into cavity theory, you will see a factor 'd' in the calculations that controls for the relative weighting of the stopping power as opposed to the mass attenuation coefficients.  The Attix textbook should be a good reference.

Dr. M.D.Belley is very precise regarding, let's say, the "typical conditions". As far as I know, several Monte Carlo codes employ also stopping power data in the corresponding simulation of the radiation transport. That is because, after the interaction of a x-ray or gamma photon with matter, an electron or a positron may be released. This thereafter interacts with matter, therefore there is a need for both attenuation-absoption coefficient data and stopping power data.

As the term implies, stopping power refers to the linear energy loss per differential distance. However, this can be also viewed for the x-rays, gamma-rays. Simply it is not the practice. Another issues has to do with the measurement of stopping power, which, usually, employs chamber detectors, which have a cavity in them.

An advantage is that there are different available data for the attenuation-absorption coefficients of phortons, namely NIST-EPDL-ENDIF-StormIsrael data  with differentiations among them. Another advantage is that the coefficients can be converted to cm2/g which are easier to handle when irradiating areas because they account for density differentiations.

As a complet of above answers, the using of stopping power or mass absorption coefficient depends on precision of calculation.  In the photon intraction with matter if some of produced electrons have energy more than the cavity radius, you should use the stopping power for just these electrons (burlin theory).

I would refer you to my bible: The Physics of Radiology (Johns & Cunningham) chapters 6 and 7. As a physicist approching radiation therapy, I was sure to have fully understood the meaning of radiation dose only after digesting the above J&C textbook.

In my opions, the stopping power (j/m) is similar to LET( linear energy transfer), which is critical parameter for the raidation biology results,usually the RBE was used as index, for example, when the proton with LET about 25Kev/μm is desired , when the RBE reached maximum valule.

The mass energy absorption coefficient is only a physical parameter. for example , the same absord dose for neutron and photon has difference biology results

Charged-particle stopping power, dE/dx, is the last link in the chain of energy absorption in matter, by external radiation, whether by photons or any by other incident radiation, whatever the energy.

In the range of low photon energy radiation (up tp 300 keV) the primary standard to measure absorbed dose is the Free Air Ionization Chamber. For higher energy, the primary standard is the Cavity Chamber. The range of the secondary electrons becomes higher than the photons as the energy increases. Therefore, the Free Air ionization chamber which lies in the principle of the electronic equilibrium is used to measured dose up 300 keV.  The cavity chamber is based on the relationship between the dose in chamber’s wall and its gas. Therefore, it has been not possible to design one cavity chamber with a very tine wall to measure low photons energy. In summary, in the range of low energy, the determination of absorbed dose is you use based on the electronic equilibrium.  In the range of higher energy, it is based on the Bragg-Gray principle~.