I think the main geometric point is that there is a difference between conformal maps (which preserve the angles between tangent vectors) and isometries (which also preserve all notions of distance -- and in particular, norms of tangent vectors).
As people have said, the Möbius group tells you what the holomorphic (complex) automorphisms are of the sphere, $S^2 = \mathbb{C} P^1$. But in one complex dimension (i.e., for Riemann surfaces) you can think of orientation-preserving, conformal (i.e., angle-preserving) transformations as the same thing as complex transformations.
So the only difference between Möbius transformations and isometries are, really, that (1) the former has to preserve orientation, and (2) the former need not preserve distances -- only angles. This explains why, for instance, scaling is a Möbius transformation, but is not an isometry.