Not sure if you get the C rate thing yet.. its a measure of how fast you're moving the electricity in and out of the battery pack. So for a 10ah pack, if you pull 10 amps out of it continuously, that is 1c, and the same for charging at that rate. 20ah at 20amps draw, same 1c rate. That is about what a regular lead battery can withstand. So to sustain say a 30amp draw, you would need a 30ah pack, or you're going to be working it too hard for long life. With many lipos, they have a 30 or 40 c rating. so for a 30 amp draw, you would only need a 1 amp hour pack, if you go by the rating, but maybe you don't even want that c rate for long life. It would be silly to have a pack that small of course, so your normal size lipo pack can handle hundreds of amps at peak without damage.
So its not that a lipo will run longer or discharge deeper, an amp hour is an amp hour, but you can put a massive amp draw on a smaller pack. But if you're hitting a lead battery with a 2 or 3c amp draw, it will maybe only deliver half of the rated amp hours, and it will die an early death.