These batteries can provide more current than the older type batteries, and sometimes your lights are current limited by the battery. It won't make them last longer though if you use more current to get more light. Let's say your light can draw 3 amps at 3.7v, but your battery can only provide 2 amps, then the light will be limited to what it can get. At 2 amps with a 2600maH battery you'll get just over an hour of use. With a battery that can supply 3 amps, at 3500maH, you'll get about the same runtime but a much brighter light.

You can buy cells that were developed for use in a Tesla, or you can buy cells that were developed for use in a cheap laptop. Which do you think will be better? I'd go with Elon Musk, even though I think he's a huckster. Whether they are better for the light you own you'll have to find out for yourself.