A big part of the difference in range between small and large aircraft is simply mathematical.
When you make a 3D object larger, area increases with the square of length while volume increases with the cube.
The A320 has a wingspan of 35.8m, carries 150 (2-class), and carries 24,210L of fuel (standard).
The A380 has a wingspan of 79.75m, about twice the dimensions of the A320.
However, the passenger capacity increases by a factor of 4 to 644 (2-class), and the fuel capacity increases by over a factor of 10 to 320,000L. There is simply more room for fuel in a larger aircraft.
Smaller aircraft may have options to carry additional fuel talks to increase range.
Note that this trend also applies to business jets despite their market having less of a correlation between range and passenger demand.
In addition to the geometrical constraints, airlines have other reasons for not operating large planes on short routes , or vice versa, more often.
Using small planes allows for more frequent service and more flexibility even if there is enough demand to use a smaller number of larger aircraft.
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There are five points that need to be addressed on the graph of range vs capacity, not just two.
Short and thin.
These routes are flown by small aircraft such as puddlejumpers and regional jets as well as smaller mainline aircraft, and represent the spokes in the classical hub-and-spoke model (Savannah-Atlanta or Bordeaux-Paris for instance).
Service frequencies are low to modest, and competition is generally fairly limited, with one or a few carriers on the route.
Short and fat.
These routes mainly exist in areas where there are large cities close together, but land/sea connectivity has a tough time competing due to geographical or political issues.
Examples of this include Tokyo-Osaka and Taiwan-Hong Kong.
Oftentimes, service frequencies are high, competition is fierce, and both narrowbody and densely configured widebody (2-class seating at best)
aircraft can be found.
Long and fat.
These are your classical, long-haul hub-to-hub routes such as London-New York or Los Angeles-Sydney.
Most of these flights are on traditional 3-class or 4-class configured widebody aircraft, and see relatively infrequent service compared to shorter routes.
Long and thin.
These routes are introduced to relieve congestion from major hubs, usually in international markets.
An example would be Newark-Glasgow, bypassing the otherwise-obligatory plane change at Heathrow or Amsterdam.
Other examples of this pattern include US domestic long-hauls.
In the middle.
Most short-to-intermediate haul mainline flying is this way -- these may be busy hub-to-spoke connections, or inter-hub flights that aren't fat enough or long enough to sustain the use of larger aircraft.
LCC networks (Southwest, JetBlue, Ryanair, Easyjet) center around these types of routes because they work well for a homogenous fleet (vs. hub and spoke networks, which require smaller planes for the spokes and bigger planes for hub-to-hub).
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Basically the airlines need keep their planes always flying, parking them idle on the airport just costs money.
Parking a big plane and waiting a complete day until it gets full with passengers is expensive, flying a small plane a long time while passengers already waiting for the next flight is lost money.
In the past big planes were build for short and medium-routes, like the B747-100SR/300-SR or the A300 (first time
two aisles one
a twin-engine?).
The 747-SR was used a lot inside Japan , while the A300 founds it first customers inside USA.
But the airlines don't buy them anymore.
Consider that on short routes you are often connecting smaller airports with fewer passengers per trip, therefore a smaller plane is more efficient.
On the other side, most jet-engines become more efficient when they are bigger, furthermore long boarding times are not a problem on long distance flights.
Also, one, two or three flights during a day on long routes are okay for passengers on long routes (selecting a round about departure/arrival), while on short routes you may need a flight every hour (selecting exact departure/arrival).
Notable exceptions are current B737-700ER and A319LR, both can fly around 10.000 km.
I don't know the sales figures, but I think only customers with special needs buy them.
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There are two design reasons for this: Airline routes In the hub-and-spoke system most large airlines use, larger volumes of passengers are concentrated into a few airport hubs then moved greater distances between hubs.
In that system, there is little need for long distance/small passenger aircraft.
Fuel Capacity
A larger plane can carry more fuel and thus go farther.
Yes, you can get a 737 configured for business that can go as far as an early 747 but, when loaded like that, it cannot carry very many people.
A BBJ 737 with full fuel can only carry about 2500 lbs—about 15 passengers with no bags, no food, no nothing.
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In general as an aircraft gets bigger, with more passengers, the bigger the wing has to be to carry all of that extra load.
As the wing gets longer, the root has to increase in thickness quite a bit due to the increased bending moment (also due to the increased load going into the wing).
As the wingspan gets longer, the wing chord usually increases at the root as well (on commercial aircraft).
Finally, since the wing root has has gotten thicker and longer, the center wing under the fuselage has also gotten taller and longer.
Considering the vast majority of fuel is stored in the wings and center wing on commercial aircraft (on many that is the only location of storage), and nearly all of the wing between the spars is dedicated to fuel storage, a higher capacity aircraft just has far more fuel storage volume available.
Of course you could put a huge wing on a small aircraft to give it a lot of range, but I think the other answers have done a good job of explaining why that isn't done.
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Why we rarely see an A320 (or equivalent aircraft) on a long route with all premium business seats or an A380 (or equivalent aircraft) with high density seating and on a short route?  On top of that, it is a much easier task to design a long range light aircraft and a short range heavy/high capacity aircraft, so why have no airliners (as far as I know) ever been designed with these characteristics in mind?
Why there is such a strong correlation between the range of an airliner and its passenger capacity?
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Using small planes allows for more frequent service and more flexibility even if there is enough demand to use a smaller number of larger aircraft. In other words, 2 or 3 flights a day on longhaul routes is OK but shorter routes need more frequent flight. Moreover, long boarding times are not an issue for long distance flights but would be on short flights. Planes make money in the air not on the ground so keeping a big plane on the ground until it is full is losing money. Finally, large planes can carry more fuel and thus go further. The main reason this happens is for efficiency and flexibility. It helps to use small planes to have shorter boarding times and more frequent flights for short trips that usually have less passengers, while for longer flights it helps to concentrate passengers in a bigger plane, since longer boarding times and less frequent flights are to be expected in these types of trips. What is more, the airlines do not profit from keeping their planes idle so they schedule their flights accordingly. Another factor is that the bigger a jet-engine is, the more efficient it is and bigger planes have more fuel capacity, thus they can cover longer distances.
The main reason this happens is for efficiency and flexibility. It helps to use small planes to have shorter boarding times and more frequent flights for short trips that usually have less passengers, while for longer flights it helps to concentrate passengers in a bigger plane, since longer boarding times and less frequent flights are to be expected in these types of trips. The airlines do not profit from keeping their planes idle so they schedule their flights accordingly. The bigger a jet-engine is, the more efficient it is. Also, bigger planes have more fuel capacity, thus they can cover longer distances.