R0 tells you how many others each person infects on average. So R0 is in one sense the measure of contagiousness—it just tells you how contagious people with the disease are on average.
Consider two different diseases with the same R0, let’s say R0 = 2. So each person on average infects 2 others. For the first disease, almost all patients infect exactly two others, but for the second, plenty infect two, many infect one, and a much smaller number infect 10 or even more others. So the average is the same, but the distribution is very different.
Given some other assumptions, this paper shows that diseases more like disease one will end up infecting many more people in the end than diseases like disease two, even though they have the same R0. So it is important to understand the distribution of secondary infections in addition to the average when predicting the final outbreak size. Contact tracing (seeing who people with the disease came in contact with and checking to see whether they end up getting infected) allows epidemiologists to do that.
Specifically, this is known as a hubness effect (when the distribution of the number of times an item is one of the k nearest neighbors of other items becomes increasingly right skewed as the number of dimensions increases) and (with certain assumptions) should be related to the phenomenon of these being closer to the centroid.