He's not talking about a NiCd or NIMH, both of which have NOTHING to do with how a LiPo performs.
The answer to your question is YES, without a shadow of a doubt. Case and point - my rifle was 400fps and I was running it with a 7.4v 15C 1600mah LiPo
With NiMH batteries, they inherently can't supply all the amperage that is needed to run an AEG motor, so to make our guns faster we increase the voltage.
With a LiPo, all the amperage you need is already there, so you can have the same rate of fire with less voltage.
LiPo batts also have a higher energy density than NiMH, which means more Mah in the same amount of space. They're also (in some cases) significantly less expensive
8.4v 5000mah elite battery - ~$100
11.1v 5900mah 20C LiPo - ~$40
As for drain, it's a matter of the math. Either way it's not significantly different from a NiMH, and again, you can have more Mah with a LiPo in the same space a NiMH battery fills.
If you opt for a simple mosfet switch instead of a triggermaster, you might even be able to find a 7.4v 2200mah LiPo that fits in an AN/PEQ-15, and that would definitely last longer than a 1500mah NiMH