9V battery versus 6xAA batteries
Let's compare using the figures from this website. I've made actual measurements in the past, and their numbers look plausible enough. 



6AA batteries in series will, according to the numbers above, have an internal resistance of 0.9 ohm. 

The servo shown at the beginning of your video, the SG-5010, has a current draw of 550mA according to web sources (probably stall- which is start-up current is higher by perhaps 50% or more). So let's say 1A to be safe. 

So with a 9V carbon-zinc battery, the battery simply cannot supply 1A. The short-circuit current is less than 260mA. So clearly it will just effectively short the supply and fail to move. 

The 9V alkaline cell with 1.5 ohm internal resistance can supply 2A with a 3V drop (to 6V) so it will almost surely work when fresh. 

The 6x AA batteries with 0.9 ohm internal resistance can supply 3.3A with a 3V drop so it also will work. 

Note that the way batteries work is that the internal resistance increases as the battery is depleted, while the open-circuit voltage stays fairly constant until the battery is about completely dead. 



The above is the analysis that corresponds to your question, however the other question is regarding the energy storage and at what point the battery will be depleted to the point where it will no longer work. 

We can ignore the carbon-zinc 9V since it won't work at all. Here is a good quality 9V alkaline cell, and looking at the curves (and extrapolating) maybe 1 hour at 300mA is possible. 

Using this datasheet for a good quality alkaline AA cell, it looks more like 5 hours operation at a constant 300mA. In practice the AA cells will also be able to start the motor further into their discharge as well, because of the lower initial internal resistance.