$\newcommand{\HH}{\mathbb{H}}$Here is an expansion of what Anton is saying.

Suppose that $M$ is a closed hyperbolic three-manifold. It follows that the universal cover of $M$ is $\HH^3$: hyperbolic space. The covering map of $M$ comes with a deck group - namely there is an action of $\pi_1(M)$ on $\HH^3$ so that the quotient $\pi_1(M) \backslash \HH^3$ is homeomorphic to $M$.

Now fix a set $S$ of generators for $\pi_1(M)$. Let $\Gamma = \Gamma(M, S)$ be the Cayley graph for $\pi_1(M)$ relative to $S$. Also, fix a point $x$ of $\HH^3$. Let $\rho_x{:}\,\Gamma \to \HH^3$ be the orbit map: namely $\rho_x(g) = g \cdot x$ for vertices of $\Gamma$, and all edges of $\Gamma$ are sent to geodesics. The Švarc-Milnor lemma says that $\rho_x$ is a quasi-isometry.

It follows that the Cayley graphs for any pair of closed hyperbolic three-manifolds are quasi-isometric. Thus homeomorphism types cannot be distinguished this way.