The Functional-Calculus Version of the Spectral Theorem In the book Analysis Now by Pedersen, the Spectral Theorem is that, for a normal operator $T$ acting on a Hilbert space $H$, there is an isometric star-isomorphism between $C(\text{sp}(T))$ and the $C^*$-algebra that is generated by $I$ and $T$. This star-isomorphism is called the continous functional calculus for $T$. 
I am under the impression that this is the first -- or at least an early -- version of the Spectral Theorem (for the infinite-dimensional setting). First, what does this tell us, that is, why would one care about a functional calculus? Second, how does this relate to the more common multiplication-version of the the Spectral Theorem?

The first spectral theorem was von Neumann's representation of a self-adjoint operator $A$
$$
             Ax = \int_{-\infty}^{\infty} \lambda dE(\lambda)x
$$
where $E(\lambda)$ is a non-decreasing orthogonal projection-valued function of $t$ on $\mathbb{R}$ and $x\in\mathcal{D}(A)$.
Earlier specialized versions were aimed at understanding the integral and discrete eigenfunction expansions associated with Sturm-Liouville ODE problems, and with the partial differential equations that gave rise to the Sturm-Liouville problems through separation of variables.
