Basically for the same reason that the British can do jokes about Hitler, but if a bunch of Germans said 'Heil Hitler' on television, even if it was a comedy segment, it would be very inappropriate.

Racism is still a huge factor in America. People forget this, but the civil war only ended in 1865. The last slave died in 1971. A lot of black people in America actually met a slave, and almost all of them know someone who did. The civil rights movement has been ongoing for decades, with many people who were teenagers in the 1950s being grandparents today, and many who survived the race riots first hand still raising their own kids.

Economic opportunity and cultural differences remain problematic. America has elected their first black President, but it's not a feat that's likely to be repeated any time in the near future. While young black people have similar opportunities to other people in their socioeconomic class today, those classes take generations to navigate, leaving the bulk of African Americans in poverty, with few having managed to claw their way far up from slavery.

African Americans use the N-word in their music because they are entering a generation where their elders heard it constantly, and their kids hear it rarely. The word itself is a cultural touchstone, reminding them of their shared background of racial oppression. Even if that oppression is over with (and that's a big and complicated if I won't take a side on today), it's legacy is powerful enough that the word itself is vital to the continuity of African American culture, at least for the foreseeable future.

I hope one day we can live in a world where slavery and racism are such distant memories that they don't affect us any more than Julius Caesar or Charlemagne do today. Then, the n-word will be as irrelevant as jokes about 'barbarian' Germans or 'viking' Swedes are today. But that day has not yet come.