To add a more general, optimistic answer to the question in the title, rather than the question in the question --- while the isomorphism class of $\pi_1(M)$ is not determined by the Laplace spectrum, it is indeed possible to extract topological information.

Let $M$ be a $d$-dimensional Riemannian manifold with Laplace spectrum $\lambda_k$. McKean and Singer showed that the heat trace $Z(t) = \sum_{k=0}^\infty e^{-\lambda_k t}$ has the short-time asymptotic expansion
$$ (4\pi t)^{d/2}Z(t) = \operatorname{Vol}(M) + \frac{t}{3} \int_M (\mbox{scalar curvature}) + \frac{t^2}{180}\int_M (10A - B + 2C) + o(t^3) $$
where $A$, $B$, and $C$ are polynomials in the curvature tensor. They observe that in the case of $d=2$, by Gauss-Bonnet, the second coefficient is a multiple of the Euler characteristic. I am not sure what one can say about higher-dimensional manifolds using Chern-Gauss-Bonnet.

As a second example, Cheeger and Muller independently proved that the analytic torsion of $M$ is equal to its Reidemeister torsion. The analytic torsion is the zeta-regularized determinant of the Laplacian acting on differential forms, while the Reidemeister torsion is defined in terms of a unimodular representation of $\pi_1$ and twisted homology. (Sorry I'm not more precise, I haven't thought about this in some time. Liviu Nicolaescu has good notes on this subject, which I recall were helpful to me.)

For further reading, check out Chavel's lovely book *Eigenvalues in Riemannian Geometry*.