Start with the group algebra $\mathbb{C} G$, which I understand to be finite formal $\mathbb{C} $-linear combinations of elements of $G$ that combine under multiplication in the obvious way from the group law.
One can then consider $l^2(G)$ which is the same as above except now the linear combinations can be infinite as long as the coefficients are square-summable. Each element of $G$ is now an isometric operator on this Hilbert space, given by left multiplication. These elements generate an algebra of isometries.
One can take the norm closure of this in the space of bounded operators on $l^2(G)$ to get what is called the "reduced C*algebra" of the group.
Or one can take the weak-* closure and get the Von Neumann algebra $L(G) $.
I am trying to understand the difference between these two operations from a beginners perspective. One question is, what is an example of an operator that you can get from the weak-* closure but not from the norm closure? Also, why is the Von Neumann algebra more interesting/more studied?
First, I don't agree that the $L(G)$ is either more interesting or more studied thatn $C_r(G)$. Both play their roles. What is true is that in a von Neumann algebra you have more tools available because of the abundance of projections.
For an example of one of the more studied of these guys, the C$^*$-algebra $C_r(\mathbb F_2)$ is projectionless, while $L(\mathbb F_2)$ is the norm closure of the span of its projections. Among lots of things, having projections (and a faithful trace, but the C$^*$-algebra already has that) allows Voiculescu's formula: $$ L(\mathbb F_n)\simeq L(\mathbb F_{(n-1)k^2+1})\otimes M_k(\mathbb C). $$ In particular, $$ L(\mathbb F_2)\simeq L(\mathbb F_5)\otimes M_2(\mathbb C). $$ This was further independently generalized by Dykema/Radulescu and their interpolated free group factors.
Note that the above isomorphisms couldn't hold at the level of the reduced C$^*$-algebras, because $L(\mathbb F_2)$ is projectionless, while $L(\mathbb F_5)\otimes M_2(\mathbb C)$ contains the non-trivial projection $$ I\otimes\begin{bmatrix}1&0\\0&0\end{bmatrix}. $$