The LED is just the LED. If the drop-in only had an LED and the reflector then you have to use a battery source around 3.5-3.7v. Higher and the LED might draw too much current and overheat, lower and the LED might not be bright enough. Because of this the LED is coupled with a driver, in this case a buck driver which lowers the voltage to something the LED accepts for a certain current.
Will running the light at 9v be brighter then 6v, well that depends a lot on the driver used but from what I read about those drop-ins, anything above 5-6v is pretty stable output wise(as in 6v will be similar in output to 9v).
You should get more runtime with 9v but output should be pretty similar.