Maximum range flying is done so we can fly the maximum distance for the amount of fuel onboard. This can be set against distance covered through air or over ground. It is of interest to a pilot when he wants to fly with the least amount of fuel flow for a distance traveled to save fuel. Maximum endurance flying is done when the pilot wants to remain airborne for the maximum time possible given an amount of fuel.
Both are influenced by engine settings, altitude flown, aircraft weight and more as we will see shortly.
Endurance is obtained when using the least amount of fuel to maintain a given altitude, so in this case time is important and not distance as with range. Which happens when holding for weather or traffic to clear.
The speed used for flying endurance is at the bottom of the power required curve, that's almost VX. This is the speed where the aircraft has the minimum sink speed during glide and it is also known as the best glide speed, Vglide. Using minimum power to maintain altitude translates in minimum fuel flow and therefore the maximum time aloft.
Maximum endurance is flown at the lowest practical altitude, because endurance is flown at an IAS for a given weight. And as TAS increases (with the same IAS) with height and power = drag x TAS you will need more power (fuel) at a higher altitude thus reducing endurance. Conclusion: stay low for endurance flying.
More weight means more power required and a higher angle of attack and drag. Minimum speed increases with a weight increase so endurance suffers. So the message is simple: loose the weight or loose the endurance.
It is a well known fact to aviators that it costs fuel to carry fuel. Thus by always flying with full tanks, more fuel is used by the engine to climb and cruise aloft, which is not very efficient in the end. Certainly commercial aircraft must carry as much payload as possible and enough fuel to perform the flight safely with all precautions taken without taking too much fuel onboard.
Calculate your fuel onboard for the trip by taking into account: startup, taxi and climb fuel (about 15 min of cruise flying), trip fuel, 30 min legal minimum (45 min would be better) and at least 10 to 15% contingency fuel and do not forget the extra fuel should you need to divert.
Make sure that you always lean the engine when flying any level altitude and not only when above 3000 ft as instructed in some countries. Not leaning properly wastes a lot of fuel and money at the same time.
It implies flying from one place to another with the highest TAS in a no-wind situation or the highest GS when wind components are added. It also means that efficiency will not be at its best because we want to get somewhere in the minimum amount of time and not with minimum amount of fuel used.
Minimum flight time also implies that the engine is operated at its maximum power setting so that maximum TAS can be achieved, thus reducing SFC.