KORD airport for instance charges domestic vs international arrivals differently.
I could see that this may have something to do with imports/taxes/tariffs etc, but why are the landing fees measured in \$ per 1,000lbs?
From Anne Graham's book, Managing Airports:
This charging mechanism uses the ‘ability to pay’ principles, since airlines using larger aircraft are in a better position to pay higher charges. Some costs such as runway wear and tear do increase with weight and also larger aircraft require vortex separations, which can reduce the number of aircraft movements during a certain period. Overall, however, there is not a strong relationship between aircraft weight and airfield cost. A flat rate landing charge for all aircraft types may be more appropriate, particularly at congested airports. This is because the cost of occupying the congested runway is movement related and independent of aircraft size. Each aircraft movement will consume the same resource.
The wear and tear of the runway depends very strongly on the weight of aircrafts, so I expect it to be at least taken into account. By discussing the physics of this below, I will show that the effect is stronger than intuition may commonly suggest.
It is certainly not the only factor to determine the fee, and may even be a minor one in most cases, but it could be of practical importance for more uncommon cases, (like when a runway is already in bad shape in some way).
I'll do a rough estimate of the incremental damage caused by a plane, on average, in comparison to planes of different weight.
The wear of the runway and taxiways depend on the load per area that is caused by the wheels. Take note that heavier planes use larger wheels to reduce that load per area.
Regarding the basic physics, the load per area is comparable to the axle load of cars. For the street wear caused by a car, this article  states that it is proportional to the axle load raised to the fourth power.
That means a double load per area causes 16 times the wear, or four times load per area 256 times the wear.
Even without an estimate of how much reduction of the load per area is provided by using larger wheels, I think that means that, for all practical purposes, the wear is caused only by the "large" airplanes.
: "Die Kosten der Verkehrsinfrastruktur" Page 12 (German, PDF) refering to work of the american State Highway and Transportation Officials (AASHO) during the 1950ies, see Wikipedia (German) on axle load.