What is the difference between reversible and irreversible adiabatic expansion? What is the difference between reversible and irreversible adiabatic expansion?
Is it true that the work done by the gas is the same but the pressure applied externally differ between two process?
If $P_{ext}$ = work done by gas, then how will there be an expansion, if they are pushing against each other at the same force?
 A: 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).


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*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.
A: There are 3 questions in your post:
1) What is the difference between reversible and irreversible adiabatic expansion?
Answer: Reversible means actually the time-reversible.  When system pressure is applied against the external surrounding (resistant), the external pressure reacts back at the next instant. The system pressure is always slightly higher than the external pressure. The other aspect of reversible is the external pressure gradually rises; whereas in the irreversible condition, the external pressure is constant.
2) Is it true that the work done by the gas is the same but the pressure applied externally differ between two process?
Answer: No. Workdone in reversible is always more than the workdone in irreversible.  The reason is the system has to work against the gradual higher external pressure in reversible condition. Whereas the external pressure is constant, no rising in irreversible condition, hence work is less.  The average external pressure in reversible is in-between P-initial and P-final whereas the external pressure is P-final in irreversible process.    
3) If Pext = work done by gas, then how will there be an expansion if they are pushing against each other at the same force?
Answer: The pressure of gas during expansion is not equal but near-extra, a slightly higher.  The pressure energy (mechanical/potential) is very sensitive compare to temperature energy (thermal).  The work is performed with slight pressure difference.  It's a quasi-equilibrium phenomena. 
