What is the difference between non-adiabatic coupling and diabatic coupling?
The difference is in the representation: adiabatic representation, being exact, allows you to make adiabatic approximation. If you do not want to have adiabatic approximation, there will be off-diagonal matrix elements in kinetic energy, so the coupling goes through kinetic energy. In diabatic representation, the kinetic energy is diagonal, and the couplings come from potential.
As was pointed out above, diabatic coupling allows you to apply Landau-Zener rule and estimate probabilities of transition.