The above comments are still mostly valid with $\epsilon$-isometries:

if $(M,g)$ and $(N,g)$ are Riemannian manifolds with diameters less than $D$, then

$f:M\to N$, $x\mapsto n_0$ for some $n_0\in N$ has

$$
|d_N(f(x),f(x')) - d_M(x,x')| = d_M(x,x') \leq D
$$

and for all $y \in N$, because
$$
d_N(y,n_0) \leq D
$$

this $f$ is a $D$-isometry. So, just having a $\epsilon$-isometry for some large $\epsilon$ should not be seen as an unlikely occurrence (however, the (even weaker) notion of quasi-isometry that you used before *is* an interesting idea, but some sort noncompactness plays a big role in things, as should be obvious from all of these answers.

On the other hand, $\epsilon$-isometries and the spectrum are certainly related in some ways, particularly with lower Ricci bounds:

For example in the paper by Cheeger and Colding, "On the structure of spaces with Ricci curvature bounded below. III." (J. Differential Geom., 54(1):37–74, 2000.) they prove that for manifolds with appropriate Ricci lower bounds, under Gromov-Hausdorff convergence the spectrum and eigenfunctions converge in some sense.

For example, their Theorem 7.11 says:

For $M_1^n$, $M_2^n$, Riemannian manifolds satisfying
$$ Ric \geq -(n-1) $$
and
$$ diam(N_1^n) \leq d < \infty $$
Then for all $N < \infty$ and $\epsilon > 0$ there is a $\delta(n,d,\epsilon,N) > 0$ such that if
$$
d_{GH}(M_1^n,M_2^n) < \delta
$$
then for $j\leq N$, we have that $|\lambda_{j,1} - \lambda_{j,2}| <\epsilon$, where $\lambda_{i,k}$ is the $i$-th eigenvalue on the $M_k$.

You can get more information about this here, which also links to some interesting looking work by Lott about how the same question with the laplacian on $p$-forms.

Just to clarify how $d_{GH}$ is related to $\epsilon$-isometries, in case it is unclear. It is a theorem that the Gromov-Hausdorff distance is $<\epsilon$ if there exists an $\epsilon/2$-isometry and vice versa. A good place to read about this is here

Thus, this should give you some sort of condition on how close two compact manifolds can be in the Gromov-Hausdorff topology, if you know their spectrum. Rescale them so that $Ric \geq -(n-1)$, and then applying Theorem 7.11 you can get a lower bound on $\epsilon$. I'm not sure how explicit the $\delta$ is in their proof, however, if this is important to you.