The traditional way of proving Grothendieck duality is to first show it for proper maps and for open immersions, which already is quite a labour. Then one uses that any morphism of Noetherian schemes factors into such and pastes the partial results together. This requires an awful lot of non-trivial checking that certain diagrams are commutative. The extension of the result to Non-Noetherian schemes then requires yet more work.
In contrast, Neeman's proof of Grothendieck duality via Brown representability is slick, short (30 pages) and conceptual and a pure pleasure to read.
(but: the first approach gives you more insight into what the functors from Grothendieck duality actually do, so it is by no means worthless)
