No one said they are the same thing. One needs both the average and the distribution to calculate the level of poverty (be it absolute or relative), and you ignored one of these.

Besides, HDI includes wealth distribution.

Nope. The sole figure in HDI related to wealth (but not equivalent with wealth, ask the Irish) is per capita GDP, which is an average. Check the formula.

_{*Lunatic*, n.
One whose delusions are out of fashion.}