The question of a dive operator requiring AOW v. number of dives v. an evaluation of the kind of background the diver has before allowing the diver to go on a more challenging dive for a particular comes up every other month or so on ScubaBoard. I will try to make a quick summary of the differences between the three to explain why the AOW is usually the preferred method.
Why have the Rule? If a diver goes on a dive that goes beyond a basic level of difficulty, there is always a possibility of an incident leading to a fatality. In that case, the operator may well be sued for allowing the diver to do a dive for which he or she is not qualified. The operator's insurance company will want to make sure they do not have to have a payout, so they will be looking for some objective sign of diver competence.
Number of Logged Dives: Tursiops pointed out that you can have a lot of logged dives without a lot of advanced experience. You can also have a lot of logged dives without any advanced experience at all. If someone who wants to do a particular dive realizes his log is 5 dives short, well, he can have 5 more logged dives in a matter of minutes. What about computer logs? In the past few years I have sold two old computers to friends. Each now has an impressive history of logged dives they have not actually done.
Evaluation of the Quality of the Diver's Experience: How will you make that judgment--on the quality of the log book described above? No matter how you do it, your dive leader is making a judgment that can later be challenged in court. It is by no means objective. This is done in places though, particularly liveaboard trips on which the diver will be doing 20-30 dives. With that much time, they have the luxury of doing a checkout dive. Operators do not have that kind of opportunity in the case of typical two tank dives.
AOW card: It is purely objective--does the diver have the card or not? It is pretty hard to fake, and it is usually easier to get the card than to create a fake. There is no operator judgment to be challenged in court. If the diver is not prepared well enough to do the trip and has an AOW card, it makes more sense to sue the instructor who gave the card than the operator who accepted it.