Examples of non Quillen-equivalent model categories having equivalent homotopy categories
The ones which usually come up to my mind as soon as I think of this:
The categories of modules over $\mathbb Z/p^2$ and $\mathbb F_p[\epsilon]/(\epsilon^2)$, $p$ a prime integer. These rings are quasi-Frobenius, so their module categories have a model structure where cofibrations are monomorphisms, fibrations are epimorphisms and weak equivalences are homomorphisms which become isomorphisms in the stable module category, obtained from the module category by killing projective-injective objects. The homotopy category is this stable module category, which in both cases is the category of $\mathbb F_p$-vector spaces. This example is equivalent to Rasmus'.
The category of DG-modules over $\mathbb F_p[v_n^{\pm1}]$, $|v_n|=2p^n-2$, $d(v_n)=0$, and the category of modules over Morava's $K(n)$.
The model category of spectra localized at $K_{(p)}$-equivalences, where $K$ is complex $K$-theory and $p$ is an odd prime, and Franke's algebraic model category $C^{(T,N)}(\mathcal A)$ defined in http://www.math.uiuc.edu/K-theory/0139/. This doesn't happen for $p=2$ by results of Constanze Roitzheim.
In all these cases the model categories are stable and the homotopy categories are not only equivalent as categories but as triangulated categories.
Daniel Dugger and Brooke Shipley give an example in their paper
A curious example of triangulated-equivalent model categories which are not Quillen equivalent
available here.