Example of groups that are not quasi-isometric but have the same growth rate? I have started working on group growth earlier this year, mainly using Drutu and Kapovich's notes. This morning I found myself wondering if I could find an example of groups that are not quasi-isometric but have the same growth rate. Spontaneously, I thought about finite groups, groups of linear growth and free groups. All those cannot provide such an example.
I firmly believe this to be possible but I have been unable to find one. Google searches have not helped me either so maybe some of you can.

To augment the answer of @QiaochuYuan, I'll turn my comment into an answer: In the realm of groups of polynomial growth, the simplest example is that both $\mathbb Z^4$ and the integer Heisenberg group have polynomial growth of degree $4$.
One still has to prove that those groups are not quasi-isometric to each other. The simplest proof that I know is that $\mathbb Z^4$ acts freely, properly and cocompactly on the topological space $\mathbb R^4$ hence its cohomological dimension is $4$; whereas the integer Heisenberg group acts freely, properly and cocompactly on $\mathbb R^3$ hence its cohomological dimension is $3$. And in the realm of groups that act freely, properly and cocompactly on Euclidean spaces (or even on contractible CW complexes), the cohomological dimension is a quasi-isometry invariant.
