What is the major advantage of the stock system over the old naming system?
The stock system is well understood world wide. And, it allows for multivalent atoms.
One advantage is that some elements have more than two common valences; therefore, the "ous" and "ic" are not sufficient to name the compounds. For example, Mn(OH)2, MnO2, Mn2O3. Another is that if the anion has more than one possible valence, the valence of the cation is called into question. An example is Na2O vs Na2O2.