Gotta know how the aircraft were supposed to be used to understand it. The -320C was developed with cargo operations in mind. All the C was, was a 320B fitted with a cargo door and a new floor.
While the C did feature new landing gear bogeies that supported a higher MTOW, this higher MTOW wasnt directly caused by the new gear. To support the cargo operations, a new cargo floor was needed. This new floor was stregthened to support a heavier payload and this is directly where the increase in payload comes from.
Now payload is everything in the plane that is being carried (pax, baggage, cargo, etc). Because of the floor area restraints of carrying passengers, the C model will have the same payload weight as a B when carrying passengers in the same configuration. However, if the C were to be carrying all cargo, it will indeed lift more weight because of the fact that cargo is volume limited , not area.
So if you look at the payload range charts, you would see that at 34,000kg, the -320C will only go ~3000nm, while the 21,000kg -320B will go over 4000nm.
Fuel flow is going to be the same because its the same exact engines. the -320B and C both have the same engines and same fuel tanks. Because of this, you are trading off fuel for payload and that is why with an increase of 13,000kg's of cargo, the -320C goes 1000nm less than the -320B.
This is the same reason why there has been no 777-200/200ER/200LR's converted into freighters yet. Boeing designed the 777 with a composite floor. When the 777-200F was designed, they replaced the composite floor beams with steel beams to support the weight of the cargo. Any converted 777's will be payload restricted simply because the floor wont be able to support the same weights the -200F does. However, structurally, the 777 airframe itself can support the same weights(relatively speaking) as the new factory built -200F's.
edit: while i say 320B or B model above, i am refering to the -320B Adv. I didnt feel like having to Adv every time i said it...