A form of PBW theorem says that the symmetric algebra and the universal algebra of a Lie algebra are isomorphic as vector spaces (in fact coalgebras and -modules). However this is not an isomorphism of algebras. One can compose the PBW isomorphism with an additional automorphism to get an isomorphism of vector spaces which restricts to isomorphism of algebras when restricted to the subalgebras of -invariant functions.
The original proof by Duflo is rather case by case, using the structure theory of Lie algebras. Kontsevich in 1998 gave a new proof which generalizes to some geometric situations in deformation quantization.