The average is a summary statistic: it takes a distribution and reduces it down to a single number. In the process we lose a lot of rich detail about the distribution. Very often this detail matters.
Consider the distribution of height of males in the UK. The average height is around 5'9", and the distribution of heights probably looks something like this:
The average works reasonably well as a summary statistic for this distribution. Knowing that the height of the average male is 5'9" lets you make a few educated guesses about the population: ~50% of people are below 5'9"; a tiny number of people are 8' or taller; a tiny number of people are 4' or shorter.
In contrast, consider this distribution of household wealth in the UK (again hypothetical, because I couldn't find data in a nice format):
The average isn't a very useful number for the second distribution. Most wealth is held by a small number of people in the long tail, and the majority of people (> 80%) have less than average wealth.