Is there an area preserving map between spheres of different radii? Does there exist a smooth map $f:(\mathbb{S}^2,r_1) \to (\mathbb{S}^2,r_2)$  which is area-preserving?
More precisely, I am considering two spheres with different radii, endowed with the round metrics. Let $\omega_1,\omega_2$ be the induced volume forms on the spheres. Is there an orienation-preserving immersion $\, f:(\mathbb{S}^2,r_1) \to (\mathbb{S}^2,r_2)$, such that $f^*\omega_2=\omega_1$?
Of course, such a map cannot be a diffeomorphism since for a volume-prserving diffeomorphism $f:M \to N$,
$$ \text{Vol}(N)=\int_N \text{Vol}_N=\int_M f^*\text{Vol}_N=\int_M \text{Vol}_M=\text{Vol}(M).$$
 A: Since $f$ is an immersion, then it must be injective, since it is then a covering map and $S^2$ is simply connected. Therefore, it must be a diffeomorphism (by the same argument on the previous answer), and we have arrived at a contradiction based on what you already know.

Previous answer (under the assumption that $\omega_1$ and $\omega_2$ were normalized volume forms).
There isn't. Your assumption implies $f$ has degree $1$. If $f$ is injective, it is a diffeomorphism and you are done (you are assuming that $f$ is an immersion: therefore, it is a submersion due to dimensional reasons and by the local form of submersions $f$ is open. It follows that $f$ is also surjective, proving that it is a diffeomorphism. You do not need to assume that $f$ is an immersion though, since you can use invariance of domain).
If $f$ is not injective, then there exists a point $y \in S^2$ such that $f^{-1}(y)$ consists of more than one point. Since $f$ has degree $1$, it follows that $y$ is a critical point, a contradiction with the fact that $f$ is an immersion.
