Nice applications of the spectral theorem?
Sorry for the necromancy, but the most basic application of the spectral theorem has to be the second derivative test in multivariable calculus, no?
If $f:\mathbb{R}^n \to \mathbb{R}$ is a smooth function, then its Hessian $D^2f$ is a symmetric bilinear form at each point of $\mathbb{R}^n$. By the spectral theorem, it has an orthogonal basis of eigenvectors. At a critical point of $f$, the definiteness of this form tells you whether the critical point is a local max, min, or saddle. By the spectral theorem, one can simply check the sign of the eigenvalues to determine this.
The Peter-Weyl Theorem. This result is, more or less, equivalent to the statement that any compact Lie group $G$ admits an injective homomorphism to some $U(n)$ (feel free to say this is still a linear algebra result, because $U(n)$ shows up). The key is that $G$ has ''enough'' finite-dimensional representations. The proof in four lines: The point is that $L^2 (G)$ is a unitary $G$-representation, but of course infinite dimensional. One finds a $G$-invariant, injective and compact operator $K:L^2 (G) \to L^2 (G)$. The eigenspaces of $K$ are finite-dimensional representations, and their sum spans $L^2 (G)$. This is not yet the result that $G$ injects into some $U(n)$, but the biggest step towards that goal.
An operator-theoretic proof that the classical Hamburger moment problem admits a solution (see e.g. Methods of modern mathematical physics by Reed and Simon, volume 2, Theorem X.4).
Weyl's proof of the Bohr analogue of Parseval's identity for almost periodic functions. More precisely, let $f$ be a uniformly almost periodic $\mathbb C$-valued function on $\mathbb R$ and let $$a(\lambda)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t)e^{-i\lambda t}dt.$$ It is known that the set $\{\lambda\in \mathbb R:\ a(\lambda)\neq 0\}$ is at most countable for any uniformly almost periodic function. Let $c_k=a(\lambda_k)\neq 0$ be the sequence of the nontrivial Fourier constants of the function $f$. Then $$\sum_{k}|c_k|^2=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}|f(t)|^2dt.$$ The proof is based on the spectral analysis of the operator $$Au=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}f(t-s)u(s)dt.$$ Weyl shows that $A$ is a normal compact operator on the space of uniformly almost periodic functions (endowed with the scalar product $(u,v)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{T}u(t-s)\overline{v(s)}dt$). The result then follows from a clever application of the spectral theorem.
A detailed exposition of the proof can be found in Theory of linear operators in Hilbert space by Akhiezer and Glazman (see Section 57).