Versions of the spectral theorem

Versions of the spectral theorem

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!