What is the difference between reversible and irreversible adiabatic expansion?
The difference is that one expansion is quasi-static (the reversible one) while the other is spontaneous because of a dramatic change of the external constraints (the irreversible one).
In the quasi-static case, you start off indeed in the state where gas pressure equates external pressure. An external operator then slightly decreases the outside pressure so that the gas expands a little bit before reaching very fast a new equilibrium state. You then repeat this procedure as many times as necessary to reach the sought pressure.
If instead you decrease instantaneously and dramatically the outside pressure (you decrease it by, say, a factor 3), then the gas will expand until it reaches equilibrium but in a fashion very very different from that of sum of minutes changes that would be done quasi-statically. As an example, for a start, quantities like internal pressure, temperature etc.. are not even well defined during an irreversible expansion.
At the end of the day, these two very different thermodynamic "trajectories" for your system will result in leading to two different final states (in your adiabatic case). And that's the reason why, the entropy change is not the same in the two cases.