Importance of universal enveloping algebras and Poincare-Birkhoff-Witt theorem.

Question

Let $L$ denote a Lie algebra and $U(L)$ denote its universal enveloping algebra. I am trying to see why universal enveloping algebra and PBW theorem are important. Precisely:

1. Given any $L$-module $V$ universal property of $U(L)$ says that $V$ is an $U(L)$-module. So what?

2. Recall that PBW theorem states that sym$(L) \cong$ gr$(U(L))$. Why is this theorem useful?

I'll be obliged if someone can explain about above things in detail.