If entropy is a state function, then why is all the talk about reversible vs. irreversible processes?

You need to consider the surroundings as well. If you go from state A to state B via a reversible process, the change in system's entropy exactly cancels out the opposite change in entropy for the surroundings; so overall there is no change in entropy. On the other hand, if it were an irreversible process, entropy change of the system (though same as the reversible case as it's a state function), doesn't cancel out the entropy change for surroundings. And overall there is a positive change in universe's entropy.

Further, in the reversible case, you can directly relate the change in system's entropy with heat transferred reversibly divided by temperature of the surrounding. But in irreversible case, the heat transferred irreversibly can't be used to evaluate entropy change and you would need to use an equivalent reversible process with same equilibrium end points.