Just like you said, the output is usually higher. Many of the lights that will "take" a 14500 push the led harder, and at any level when it is PWM type.
When it comes to total power, the watts total of a good 14500 are not that much greater than a 2000mAh rechargable. many of them are marked with capacity they dont have, or dont have for very long time of use.
It takes quite a bit more power to get a LED to be lots brighter to the human eyes. so while it might not seem that much brighter depending on the light and the battery, it can drive the LED at 2+ times the power, at least for a while when the voltage of the li-ion is high.
IN boost type lights , ya use a li-ion to get wow output, and (again) depending on the light you get it.