White privilege doesn't just amount to how people are treated. White privilege means that white people have created every single institution in America and has slanted it so it favors their culture over others. It also means that white people have been able to accrue mass amounts of wealth so that their future generations can inherit it and begin life from the start without being a financial burden (even though some families have squandered what their ancestors have earned, far more have been able to make a life of themselves than those that haven't). With these kinds of privileges specific parts of the community can afford to send their children to private school, to build nice areas for themselves and others like them to live, and with less financial instability comes a wide range of benefits that are too numerous to list here. This is just a basic overview of what amounts to white privilege, those that study it will give you a better insight if you were to read their works.
Yes at this present time the wealth disparity is more of a class thing, but white people are still at the top of the tree because of the privileges they have been able to set up for themselves, and the ones doing all the fighting to keep it that way are white people. It is so overt that you will even see it in the media, where white presenters will tell other white people to 'have more kids because all the other races will outnumber you in 20 years' type of nonsense. With those kinds of messages being relayed then the average person with little to spare will start to fear losing what little they have and they will fight to keep white people on top, whether those on top deserve it or not. Its a type of tribal mentality and we saw similar things with the African American community (at least what the white owners of TV stations would broadcast) when Obama got into the White House - as long as someone from the tribe I identify with is on top then we are all winning.
White privilege, and the tribalism on both sides, needs to disappear, and the only way to have that happen is if both sides understand it and have patience with the other until they recognize it in themselves, and I get the feeling the African American community has understood it for a long time.