As Tsuyoshi stated the issues are not as clear cut as you may think. The relation between diagonalization and relativization is more complicated. BGS shows that there are oracles that relativized to them $\mathsf{P}$ is equal to $\mathsf{NP}$ (take any \mathsf{PSpace\text{-}complete} problem), and there are oracles that relativized to them they are not equal (which is more complicated).
If you believe that they are not equal, then the easier results is what is useful. Since simple diagonalization arguments relativize (because they only use machines as black box, so the proof should work even if you replace them with their relativized version), you cannot separate them in the real world with a simple relativization (otherwise it would also separate them in relativized to a \mathsf{PSpace\text{-}complete} oracle).
BGS result can be strengthened to: no simple diagonalization can separate \mathsf{P}$ from $\mathsf{NP} (we have to define formally what it a diagonalization and when it is simple).
Informally you can think of simple diagonalization as black box separation, i.e. if the proof that there is a problem in \mathsf{NP}$ which is not in $\mathsf{P}$ does not depend on the particular code of the machine solving that $\mathsf{NP}$ problem then the proof cannot work.
(I tried to simplify and make it easy to understand, so don't take it as completely correct. If you want to understand the details see Kozen's old paper, Fortnow's survey, and a more recent paper by Impagliazzo, Kabanets, and Kolokolova).