Income/Wealth Inequality; Why does it matter?
There could be some complicated models where high inequality could have effect on efficiency - the research is inconclusive (or to be more precise almost non existent) on the matter.

However, in Public Economics, depending on the selected welfare function (ralwsian, utilitarian, charitable conservative/libertarian etc.), inequality matters for optimal taxation. For example, even in competitive market without any imperfections someone’s marginal product of work and hence wage might be close to zero or not enough to afford food. From pure efficiency perspective that would be optimal, but under Ralwsian/utilitarian or also some other welfare function there should be some transfers to said person.

This being said economics does not provide any answers into which welfare function is the right one for that you need to study philosophy. Inequality as far as currently understood is a matter of taste.