I hope no one gets offended if I summarize a couple of comments adding few details to it. Truly $U_h(\mathfrak g)$ as an associative algebra is not different from $U(\mathfrak g)[[h]]$ (here $\mathfrak g$ is semisimple, everything is char=0).
This results is just a rigidity result on the associative algebra $U(\mathfrak g)$ (which reflects rigidity of the Lie algebra $\mathfrak g$). What I wrote inside the bracket seems innocent but it is not: one may say it depends on the fact that the Hochschild cohomology of $U(\mathfrak g)$ is isomorphic to the Chevalley-Eilenberg cohomology of $\mathfrak g$. I find this is nicely explained in http://people.mpim-bonn.mpg.de/crossi/LectETHbook.pdf .
When I first saw this result (which is already in the by now classical Chari-Pressley's book) my first impression was "so what's all the fuzz about quantum groups?". The point is that:
- They are non trivial deformation of the universal enveloping algebra as a Hopf algebra.
- The isomorphism as associative algebras is neither explicit not canonical. We know it exists from purely cohomological arguments...
How does this connect to the embedding $\mathfrak g\hookrightarrow U(\mathfrak g)$? The canonical way to reconstruct $\mathfrak g$ inside $U(\mathfrak g)$ is to identify it with the set of primitive elements (primitive means $\Delta X=X\otimes 1+1\otimes X$). Therefore this embedding depends on the whole Hopf algebra structure (coproduct to determine primitive elements and product to show that they form a Lie algebra and generate a PBW basis).
The set of primitive elements in $U_h(\mathfrak g)$ is trivial and certainly does not allow to reconstruct a PBW basis.
One may look for twisted primitive elements with respect to some group-like element different from 1. This is an interesting object, it contains the analogue of simple root vectors, but its algebraic properties are rather weak. Still: starting from twisted primtive elements and performing $q$-commutators, in the context of global quantization $U_q(\mathfrak g)$ it is possible to reconstruct all "root vectors" giving a PBW basis. But there is no obvious algebraic structure even on this set of $q$-root vectors.
Of course one may try to understand some kind of embedding $\mathfrak g_h\hookrightarrow U_h(\mathfrak g)$ as a deformation of $\mathfrak g\hookrightarrow U(\mathfrak g)$ as was done in some of the mentioned reference but everything is non canonical and, in my opinion, in the long run it just turns out to be a way of building up an explicit algebra isomorphism; which is known to be technically very complicated.
(this comment does not touch on the "braided" side of the story; that, I do not understand)