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9V lasts me a long time and I do a lot of experimenting, even when I make a mistake and don't realize I connected 9V to 0V without any resistance. It takes a lot of life from it I'm sure but they do last a while even when they are drained pretty low. When they get real low I put em in series for that extra life.
As with any component or technology, it is best to learn how to find and read the engineering data for the item so that you can answer your own questions. For batteries, one must go to the manufacturer's website and find the "technical information" or "engineering data" or "industrial" link to finally get to actual battery specifications. At that point, just read the specifications for the two battery types and it will be obvious which is better.
A 9 volt battery is nothing more than 6 AAAA sized cells. Obviously 6 AA sized cells is going to win in every catagory, excluding fresh voltage, which is going to be the same for both, but last longer for the AA pack.
A 9V AC adapter's voltage stays the same as it powers something.
Look at the voltage curves of battery cells on a battery manufacturer's site to see the voltage dropping over time when they power something.
Pretty much the only thing 9 volt batteries (the common kind) are good for is smoke alarms and desktop clock backup. Sure they make quick and dirty portable power supplies for hobby circuits but they should only be used for testing.