With high inequality, money becomes more valuable and people start to think of everything in terms of money. Even romantic relationships become money-oriented. It's not uncommon for marriages to break up because of financial reasons. Wealth inequality causes more human suffering than moderate poverty.
People who lived in communist Russia will often tell you that people were happier under communism, even though they didn't have much - Often, they had to farm vegetables from their own gardens to supplement their diets. But they were happy because they knew that they had the same as everyone else. Also, people had no ulterior motives to do anything - When people liked you, they liked you because of who you were. When people did something nice, they did it because of purely altruistic reasons.
In capitalist society, money underlies the behaviour of everything done by everyone. You can never know if a person is good or bad because you never know their true intentions.
I think the media coverage of wealthy people is itself a bigger problem than inequality itself.
The media motivates people to work harder and acquire more wealth. Without media, only innately passionate people would launch startups. Today, most people who start a company do it because they want financial rewards.
It is human nature to want equality. It reminds me of a Russian joke about a man who finds a genie in a bottle. The genie tells the man that he can get anything he wishes but that his neighbour will get twice as much of whatever he gets. The man pauses to think for a moment and then says: "Take out one of my eyes."