These two metrics are not the same. The first metric is
$$ds^2 = dr^2 + r^2 d\Omega^2$$
while the second metric is
$$ds^2 = r_0^2 d\Omega^2$$
where $r_0$ is a constant, the radius of the sphere. These are different quantities; the spaces involved don't even have the same number of dimensions. The Riemann curvature of the former metric is zero, while the curvature of the latter is not.
You're correct to conclude that the first metric should have zero curvature, since it's related to Cartesian coordinates by a coordinate change and the curvature is a tensor. But this doesn't say anything about the second metric.
You might think the second case should be the same, because the sphere can be embedded into flat space. The geometrical difference is that vectors on the sphere must lie tangent to the sphere. That means that parallel transport on the sphere is not the same as parallel transport in the embedding space, since in the former case, we have to project the vector back down to the sphere after every step.