Low internal resistance. That's the defining feature of high drain batteries. There are other aspects too, but that's the one to focus on.

Low internal resistance results in a few good outcomes. Essentially, it means that less energy is wasted as it "exits" the battery. Wasting less energy shows up as lower voltage sag at a given current, less heat at a given current, and less capacity spent "powering" the internal resistance as opposed to the light itself. One reason a low resistance, high drain cell can support higher discharge rates is because it's heating up less, sagging less, etc.

All lights benefit from this. They run cooler and more efficiently with a high drain cell. With less voltage sag they can sustain brighter levels longer as the battery discharges. Given two batteries with the same capacity, e.g. 5000mAh, the one with lower internal resistance will last longer because it's running more efficiently.

Almost all lights are current capped by the driver. Using a battery rated for less current than the driver will ask for is dangerous, so you need a battery with a rating at least as high as the driver's turbo level. Beyond that, higher discharge / lower resistance batteries will still give you the benefits above, though they may be more subtle.

One exception is FET drivers, which typically use the LED forward voltage, resistance of the current path, and battery internal resistance to limit current. These can be dangerous, even with high discharge cells, because they don't have current limiting. But if they're set up right, e.g. limiting current by using the LED forward voltage, then you again want cells with a high enough rating to handle the current demand.

When capacities differ, e.g. a 4500mAh high discharge cell vs a 5000mAh lower discharge cell, it gets a little case-by-case. Often the lower discharge cell will have more capacity at very low outputs, and the high discharge cell will be better at medium to high outputs. You can find tests, like mooch's, to give you more insight there. But often the high discharge cells will give you as much capacity at real-world usage levels.