▻ AVERAGE AND TOTAL DO NOT NECESSARILY DIFFER
There is no necessary difference of result between the two criteria. Take a single society with a population of 100. The greatest total utility for this society (at a time) is 1000 utils, let's say. That's the absolute limit, the maximum total utility possible. The average utility will be 10. The average utility cannot be higher since maximum utility has been reached at 1000 - nor can it be lower. 1000 divided by 100 gives an average of 10 whatever the case.
▻ WHEN THEY DIFFER, WHAT IS THE CASE FOR PREFERRING AVERAGE TO TOTAL UTILITY?
But if you take two societies, even if they are egalitarian and each member gains the same amount of utility, it is rational to choose the society which has the highest average utility. How so? Suppose Society A has a total utility of 10,000 and 5000 members; and Society B has a total utility of 5000 and 100 members. In society A, total utility is higher but average utility is 2 utils.
In Society B, total utility is lower but average utility is higher at 50 utils.
My rational choice would be to go by average utility. As a rational agent, all else equal, I'd rather have 50 utils than 2.
▻ IS THERE NO CASE FOR PREFERRING TOTAL UTILITY ?
There is a case, especially as concerns future generations. You need to go into specialist literature here. But one argument, relating to future generations, is sketched as follows :
The total utility criterion was first applied to the study of optimal population paths by Meade (1966). A rigorous analysis in the context of a Solow-Swan model of growth was provided by Dasgupta (1969). In his model, the optimal
rate of population growth can be positive. Although the present generation may
be better off consuming all the capital stock and having no descendants, the
present gain in utility is outweighed by the loss in utility of the future generations.This result sharply contrasts with the ones obtained under the average utility criterion. ( A. Rodriguez, 'The Dependency Ratio and Optimum Population Growth: The Total Utility Case', Journal of Population Economics, Vol. 1, No. 2 (Oct., 1988), 141. Further references aplenty.)
▻ BOGUS MATHEMATICS
Talk of numerical utils is only a mode of convenience. Utility cannot be measured with any mathematical precision in real life. But we can and do make judgements of rough relative magnitude, and the above calculations are based only on this assumption.
Henry Sidgwick, The Methods of Ethics, New York: Dover, 1966/ 1907, 415.
John Rawls, A Theory of Justice, Cambridge, Mass., Harvard University Press,
1971, chapters 1 & 3.
Gregory S. Kavka, 'Rawls on Average and Total Utility', Philosophical Studies: An International Journal for Philosophy in the Analytic Tradition, Vol. 27, No. 4 (Apr., 1975), 237-253.
A. Rodriguez, 'The Dependency Ratio and Optimum Population Growth: The Total Utility Case', Journal of Population Economics, Vol. 1, No. 2 (Oct., 1988), 141-156.