As a PP stated, in a perfect world, everyone would be equal and there would be nothing but sunshine and roses and love and happiness (cue Al Green music)...
However...
Unfortunately, that is not the case. Blacks were called chimps (and in some parts of the country, we still are, even though it may be less blatant) and routinely told we were "less than" due to nothing more than our skin color. So while it is certainly not nice to call ANYONE names or animals, it is particularly cutting when a chimp reference is used in relation to a Black person. Black people were called savages and many other things (as were Native Americans - another part of my genetic makeup), and oppressed and enslaved for HUNDREDS of years. That is not something that can just "go away" or that one can "get over" overnight. In the same way that it takes time to heal from a bad accident or some other bodily trauma, it takes time to heal from the effects of slavery and jim crow.
And like other posters have said, racism is still alive, though certainly not to the extent that it was in the past. I am still followed when I enter a store, many (particularly older and southern) people find it very hard to believe that I went to college to study music and am now an opera singer, sometimes people feel the need to "explain" and "define" "big" words or concepts to me, as if I'm somehow not capable of possessing basic comprehension skills and would have no idea what they were talking about unless they "broke it down". And I've had lots of people assume that I was going to vote a certain way in the election due to the color of my skin. I've had people clutch their purses tighter to themselves when I approach. And many, many more instances that I could name...
I also agree that some people look for racism where racism doesn't exist. I think there are lots of things that may get blamed on race that are really not related to race at all. But again, it will take time to heal that sort of mentality. A good analogy might be to think of children who come from an abusive home. Once they are placed in a loving environment, it takes them some time to trust that the person/people in their new environment. They don't just "get over" everything that they've been through and begin to love their new family unconditionally.