Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras:

($*$) $A=\int_{\sigma(A)}\lambda d\nu(\lambda)$ where $A$ is a bounded self-adjoint and $d\nu$ is a $\mathcal{P(B(H))}$-valued measure.

This can be generalized to self-adjoint unbounded operators, or restricted to self-adjoint compact ones.

But I read a lot of comments in books or papers (for example p.9 here: http://arxiv.org/abs/quant-ph/0601158 ) that later it was proven that the spectral theorem is valid for any von neumann algebra $\mathfrak{M}$, with $d\nu$ a $\mathcal{P}\mathfrak{(M)}$-valued measure.

1) Since a von Neumann algebra is also a $C^*$-algebra, what new information does this "new" spectral theorem give?

2) Is the topology of convergence leading to equality in ($*$) the only difference?

3) If so can you explain why the von Neumann version is not presented as the most general one for bounded operators (since convergence in weak topology implies the norm one)?

Any further clarifications on the different versions of the spectral theorem are most welcome!