In response to the critical comments below I revised my answer. Hope this is more helpful!

**(1) Two kinds of metrics are defined on generally different spaces.**

It is not fair to compare these two metric since the Fisher-Rao is defined for probability densities defined on space $(X,\mu)$, the elements of concern are in space $M(X,\mu)$; while the Wasserstein is defined directly on probability measures on $X$ with/without densities, the elements of concern are in space $M(X)$. Even if we step back and say we concern those probability measures with absolutely continuous densities and the manifold they defined, it is readily observed that

Fisher metric provides a negative constant sectional curvature while
Wasserstein metric is flat [3]

Negative curvature is favorable ~~due to various comparison theorems in Riemanian geometry.~~(See a different opinion from Kloeckner's comment below)a strictly negative curvature can be regarded as convexity of the family of measures under concern making the minimizer of KL divergence unique. The reflection of this point in information geometry is the natural relation between KL divergence and Fisher-Rao metric, which correspondence Wasserstein distance does not have.[1] Meanwhile the Wasserstein metric has a natural connection with optimal transportation theory which Fisher-Rao metric may not provide.

**(2)What are the connections between these two metrics? They both somehow characterize the dependence between two distributions using geodesic distance w.r.t. metrics.**

*(2.1)Dependence characterization using metric geodesic distance*

(i)Fisher-Rao distance between probability densities measures is
equivalent to the geodesic distance in sense of the 'correct'
geodesic distance characterized by Cencov's theorem[6].

(ii)Wasserstein distance between probability measures is equivalent
to the most correlated copula associated with these measures. [2]

Therefore the Wasserstein metric may reflect the dependence(entropy of data-generating process) better than Fisher-Rao in strongly dependent case[2]. This statement can be made precise using Newman's language[4] describing the bounded-Lipshictz dependence mentioned in [5].

*(2.2)Fisher-Rao depends on the underlying manifold.*

However, Fisher-Rao metric also depends on the embedding manifold and have parametric and nonparametric version. For example, when the underlying manifold is all Gaussian measures with varying means, then the Rao-Fisher metric is simply a linear metric when measuring the geodesic distance between two points on this manifold; when the underlying manifold is all measures with finite second moments, then the Rao-Fisher metric is not linear anymore. So the discussion is also affected by what underlying manifold we have in mind.

*(2.3)Wasserstein metric does not depend on the underlying manifold by its definition since it must take variation over all measures with prescribed margins.*

**(3) Cencov's Theorem**

Very loosely, Fisher-Rao metric, due to Cencov theorem [6] that Fisher-Rao metric is the 'correct' metric to use when the transition mappings are selected to be Markov morphisms.

...proved that the Fisher-Rao metric is the only metric that is
invariant under mappings referred to as congruent embeddings by Markov
morphisms.[7]

**(4)What is the corresponding characterization for the Wasserstein metric?**

As far as I know there is no a characterization for general $L^p$ Wasserstein metric, but in some cases like $L^2$, the minimizer to Wasserstein metric is derived [8] as optimal couplings. These characterizations are also useful in reality [2]. As OP mentioned, the minimizer of Fisher-Rao metric can be characterized using FIsher information matrix and in exponential family these are MLEs.

**Reference**

[1]Amari, Shun-ichi. "Divergence function, information monotonicity and information geometry." Workshop on Information Theoretic Methods in Science and Engineering (WITMSE). 2009.

[2]Marti, Gautier, et al. "Optimal transport vs. Fisher-Rao distance between copulas for clustering multivariate time series." Statistical Signal Processing Workshop (SSP), 2016 IEEE. IEEE, 2016.

[3]Barbaresco, Frédéric. "Geometric radar processing based on Fréchet distance: information geometry versus optimal transport theory." Radar Symposium (IRS), 2011 Proceedings International. IEEE, 2011.

[4]Newman, Morris. "Periodicity modulo m and divisibility properties of the partition function." Transactions of the American Mathematical Society 97.2 (1960): 225-236.

[5]Bulinski, A. V. and Vronski, M. A. (1996). Statistical variant of the central limit theorem for associated random elds, Fundam. Prikl. Mat., 2, 4, pp. 999{1018 (in
Russian).

[6]Cencov, Nikolai Nikolaevich. Statistical decision rules and optimal inference. No. 53. American Mathematical Soc., 2000.

[7]Peter, Adrian, and Anand Rangarajan. "Shape analysis using the Fisher-Rao Riemannian metric: Unifying shape representation and deformation." Biomedical Imaging: Nano to Macro, 2006. 3rd IEEE International Symposium on. IEEE, 2006.

[8]Rüschendorf, L., & Rachev, S. T. (1990). A characterization of random variables with minimum L2-distance. Journal of Multivariate Analysis, 32(1), 48-54.

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