A conformal map of a Riemannian manifold $M$ with metric $g$ is a diffeomorphism $f: M\to M$ such that $f^\star g = \Omega g$ for some smooth, positive $\Omega: M \to \mathbb{R}$. If $\Omega(x) = 1\ \forall\ x \in M$, the map $f$ is called an isometry.
For example, the sphere $S_n$ admits a Lie group of conformal maps, the Mobius group, which in general are not isometries: isometries form a proper subgroup of the Mobius group.
Two Riemannian manifolds $(M, g)$, $(N, h)$ are conformally diffeomorphic if there is a diffeomorphism $\phi: M\to N$ such that $f^\star h = \Omega g$ for some smooth, positive $\Omega: M \to \mathbb{R}$. Clearly, if $f$ is a conformal map on $N$, $\phi^{-1}f\phi$ is a conformal map on $M$.
I would like to know if there are Riemannian manifolds, not conformally diffeomorphic to $S_n$, that admit conformal maps that are not isometries. For example, I think that the only conformal maps of hyperbolic space $H_n$ are isometries (although I'd be happy to see a counterexample).