Improved climb could also account for significant differences in runway length used.
Put simply if the runway length is not the limiting factor but the terrain or obstacles around the airfield, then we accelerate to a much higher speed on the runway. We then use this additional speed to allow us to climb better once airborne.
There can be 30 knots or so of additional speed at rotation. This can be quite uncomfortable on some runways where the surface is not all it might be, Moscow springs to mind.
There is therefore no simple link between weight and takeoff distance. As others have pointed out there are many factors and many companies now use laptops to produce this performance data. This can take the pilots one step further out of the loop, which in turn can contribute to other problems such as the MK 747 crash at Halifax. But that is another debate!