Because the gap indicates unfairness and people care about that. If you are poor, raising the floor will be good enough, you primary worry will be sufficient income. Once you reach a comfortable life, your priority will - or at least may - shift, now you are in a position that allows you to worry if the situation is fair.

Also narrowing the gap is a way to raise the floor including in absence of growth while raising the floor without narrowing the gap depends on growth. But I think this is not too relevant in practice because taking a lot of money from a few wealthy people will often result in rather small changes as the money has to be spread among a lot of people.