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:
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.
: "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.