f(z) = z* is certainly differentiable when viewed as a function from R^2 to R^(2). But complex differentiability is a stronger condition: it means that locally, the function looks like multiplication by a complex number. But geometrically z* is a reflection, while multiplication by a complex number is a rotation and dilatation - no complex number will give you a reflection.
However, compared to uglier functions, the conjugate is a bit of a special case - it's almost complex differentiable, it just kind of fails for an almost trivial reason. And this is why the concept of an antiholomorphic function exists: a function which is complex differentiable with respect to to z*.