can someone give me examples of the explanatory power of the quantum-mechanical model?
Power to explain what? The nature of the chemical bond? Linus Pauling wrote a book with that title. Bascially, QM can explain everything about chemistry such as structure, polarity, electronegativty, bond energy and reaction rates, and other models cannot.
The quantum mechanical model of the atom PREDICTS the four quantum numbers n, l, ml and ms (and their values).