Fairness measures the equality of the participants in the absence of some manifest distinction between them. That's why Rawls needs the veil of ignorance: to thrust lack of information into the matter of justice so that the circumstances of equality is all that can govern the judgments. Desert measures exactly the opposite: the credit or blame for what happened in terms of what was the unique contribution of each of the participants.
Fairness is about consumption whereas desert is about production. Fairness applies justice to what we should get; desert applies justice to what we already gave.
Desert applies to gratitude because gratitude is the appropriate response to giving. Fairness is not in the same domain as gratitude.
Fairness measures how your allocation feels relative to what the other guy got; desert measures how your apportionment feels relative to your responsibility for what happened, positive or negative.
As we move from civil society to mass populations we move from knowing the particularities of each person to knowing practically nothing about each person. As a result we move from an environment where justice as deservedness can function well to one where only justice as fairness can function well.