Recently I found myself a bit confused about the definition of conformality. In Riemannian geometry, we say that two metrics on a manifold $M$, $g$ and $h$ are in the same conformal class if there exists a function $\mu:M\rightarrow\mathbb{R}$ such that, \begin{equation} g = e^\mu h \tag{$\ast$}. \end{equation}
The point of confusion for me was this: the uniformization theorem says that there is only one conformal class on the sphere $\mathbb{S}^2$. This, however, does not mean that two Riemannian metrics on a sphere can be related like ($\ast$), but that there is a diffeomorphism $\varphi:\mathbb{S}^2\rightarrow\mathbb{S}^2$ such that $\varphi^*g=e^\mu h$. Why is there a difference between the definitions of conformality in complex analysis (as in the Uniformization theorem) and Riemannian geometry? What does conformal class mean in complex analysis (is it different from the definition for Riemannian metrics)?
The situation is actually (almost) the same. There are two notions in RG:
Two metrics conformal to each other (or metrics in the same conformal class), where the metrics differ by a conformal factor. Equivalently, the identity map $id: (M, g_1)\to (M,g_2)$ is conformal.
Two metrics which are conformally isomorphic, meaning the existence of a conformal diffeomorphism $f: (M, g_1)\to (M,g_2)$. (There is a minor issue here, let's assume that $M$ is oriented and $f$ is orientation-preserving.)
Similarly, in complex analysis:
Two complex structures $c_1, c_2$ on a manifold which are equal to each other, equivalently, the identity map $id: (M, c_1)\to (M,c_2)$ is conformal/biholomorphic.
Two complex structures $c_1, c_2$ on a manifold which are isomorphic/biholomorphic to each other, equivalently, there is a conformal/biholomorphic map $(M, c_1)\to (M,c_2)$.
Now, each complex structure on $M$ defines a unique conformal class of Riemannian metrics. Conversely, if $M$ is oriented, then each conformal class of Riemannian metrics defines a unique complex structure. Accordingly, if $f: (M, g_1)\to (M,g_2)$ is an orientation-preserving conformal diffeomorphism, then it is biholomorphic.
The UT for 2-d spheres simply says that there is a unique conformal class of Riemannian metrics on $S^2$ up to an orientation-preserving conformal diffeomorphism. In the language of complex analysis, UT says that there is a unique (up to a biholomorphic isomorphism) class of complex structures on $S^2$.
To conclude: The two notions in Riemannian and complex geometry (in the setting of 2-d Riemannian manifolds and Riemann surfaces) are equivalent, as long as you restrict to the category of oriented manifolds and orientation-preserving diffeomorphisms.