I would argue that the Great Society was not a success. It did not achieve its goals in any lasting way.
The two main goals of the Great Society were to end A) poverty and B) racial injustice. One can argue about whether racial injustice has been eradicated, so it might have achieved that goal. However, there is no way to argue that poverty has been eliminated in the United States.
The real problem with Johnson's initiative was that government simply cannot do away with poverty or, to a lesser extent, with racial injustice. The government can ban overt discrimination, for example, but it cannot force blacks and whites to go to school together and, thereby, ensure that blacks get a good education. There are still racial disparities in educational achievement today.
If it is hard to end racial injustice, it is much harder to end poverty. No government has yet found a way to achieve economic growth without leaving some people behind. Poverty continues to exist today in the US.
If the Great Society was meant to end racial injustice and poverty, it was a failure because both problems continue to exist today.