I have a confusion which I suppose must be rather basic.
As I understand, in the 60s/70s it was known that the empirical eigenvalue distribution of the sample covariance (of $n$ i.i.d. standard normal $p$-variate observations) converged to the Marcenko-Pastur distribution when $p/n$ converges to a constant.
On the other hand, I. Johnstone decades later established that the largest eigenvalue of this sample covariance converges to the Tracy-Widom distribution.
I am not understanding how this second result was a breakthrough from the first. Doesn't the fact that we know the limiting empirical eigenvalue distribution of the sample covariance mean we get for free the distribution of the top eigenvalue?
What am I missing?