Manifold of probability measures: connections between two types of metrics The space of probability measures could be viewed as an infinite-dimensional manifold, equipped with two possible types of metrics — (1) Wasserstein and (2) Fisher-Rao. Metric (1) is connected with optimal transport, and metric (2) is connected with information geometry. 
Question: What are the connections between these two metrics? We know that the Fisher-Rao metric is characterized by the Fisher information matrix, but what is the corresponding characterization for the Wasserstein metric? Any references are much appreciated.
 A: Just a quick follow-up: Very recently (well, actually in 2015) three teams came up independently and almost simultaneously with the same construction of a new "optimal-transport-like" distance on the space of Radon measures $\mathcal M^+$, which somehow interpolates continuously between Wasserstein and Fisher-Rao. This distance now goes by the name of Wasserstein-Fisher-Rao (WFR) metrics, sometimes also Hellinger-Kantorovich (HK) distance, and gave rise to a whole new topics generally referred to as unbalanced optimal transport. As pointed out by @GabeK in his excellent answer the Wasserstein and Fisher-Rao structures interact in an interesting way and lead to unsuspected behaviour (well, at least for me). The underlying structure possesses several rich and geometric underlying formulations. In a nutshell, the WFR distance can be seen as an infimal-convolution of Wasserstein and Fisher-Rao distances.
For recent developments see the citations of the original trhee papers below (I belong to the first team)

[1] Kondratyev, S., Monsaingeon, L., & Vorotnikov, D. (2016). A new optimal transport distance on the space of finite Radon measures. Advances in Differential Equations, 21(11/12), 1117-1164.
[2] Chizat, L., Peyré, G., Schmitzer, B., & Vialard, F. X. (2018). An interpolating distance between optimal transport and Fisher–Rao metrics. Foundations of Computational Mathematics, 18(1), 1-44.
[3] Liero, M., Mielke, A., & Savaré, G. (2018). Optimal entropy-transport problems and a new Hellinger–Kantorovich distance between positive measures. Inventiones mathematicae, 211(3), 969-1117.
A: 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.
A: Edit (June 2022): Jun Zhang and I wrote a survey paper on some interactions between these two fields which expands on what I mentioned here. You can find the paper at the following link: https://arxiv.org/abs/2206.14791
Original answer:  This is an excellent question, but I don't think there is a clean and simple answer. In general, these two metrics reflect different things for probability measures and they interact in interesting ways.
Before getting too much into the weeds, I'll mention a paper that I found extremely helpful to my own understanding of the relationship.
Khesin, B., Lenells, J., Misiołek, G., & Preston, S. C. (2013). Geometry of diffeomorphism groups, complete integrability and geometric statistics. Geometric and Functional Analysis, 23(1), 334-366.
http://www.math.toronto.edu/khesin/papers/H1-gafa.pdf
With that out of the way, I'll mention a few general phenomena:

*

*The Fisher-Rao metric is a canonical Riemannian metric on a parametrized statistical manifold. Generally, one does not do optimal transport of a finite dimensional parametrized family of distributions, so to make any comparisons, it is instructive to consider the infinite-dimensional non-parametric version of a statistical manifold. The paper that I cited above does this extremely clearly and explains how to extend the Fisher-Rao metric to the infinite-dimensional setting as a particular $H^1$ metric.
There have been other attempts to study non-parametrized statistical manifolds, but these generally involve Orlisz spaces and a lot of seemingly ad hoc machinery. To be honest, I am not aware of what new mathematical insights this more complicated theory yields. Perhaps someone with more experience can fill me in.


*It is worth noting that the space of probability measures on a metric space $X$, with the distance induced by the Wasserstein 2 metric is not, in general, a Riemmanian manifold. However, once some extra regularity has been assumed, Otto did introduce a formal Riemannian structure on this space. This is an important notion, but it is formal. The curvature of this structure depends on the underlying metric of $X$. In particular, $(P(X),\mathcal{W}_2)$ has non-negative sectional curvature (in the sense of Topogonov) if and only if $X$ does. For more information on this, the following paper of John Lott goes into a lot more detail. https://math.berkeley.edu/~lott/cmp.pdf


*By contrast, the $H^1$ metric on probability distributions does not depend on the metric, which is somehow a consequence of the fact that the entropy is diffeomorphism invariant. As shown in the first paper I cited,  the space of probability densities with $H^1$ metric is isometric to the positive part of the sphere in Hilbert space. As such, it has constant positive sectional curvature. Furthermore, this makes distances and geodesics in the $H^1$ metric much easier to compute.


*I've somewhat been dancing around the issue, but the Wasserstein distance and $H^1$ metric are defined for different spaces. Furthermore, if one considers the topologies induced by the corresponding metric, these are different. In general, the Wasserstein distance is defined for more general probability measures whereas the information-geometric notions generally require probability densities.


*There are many important theorems that relate the entropic notions of statistical distance to the optimal transport notions. For instance, the Talagrand inequality shows that the squared Wasserstein distance is bounded by the relative entropy (under assumptions). For more information, the following paper of Otto and Villani is a great reference. http://cedricvillani.org/wp-content/uploads/2012/08/014.OV-Talagrand.pdf
In general, we can expect to have inequalities that control the Wasserstein distance by the entropic distances. However, we don't really expect to have inequalities going the other way. There are multiple ways to think of this, but it boils down to the fact that the entropic distances generally require more regularity in their definition and we don't expect to be able to bound higher derivatives by lower ones.


*It seems that information geometry does not only use the Fisher metric, but also many other "divergences," which are distances where you drop the requirements of symmetry and the triangle inequality. There is a theorem that shows that a very general class of divergences (the $f$-divergences) induce the Fisher-Rao metric, which is why it is a canonical metric. However, it is often worthwhile to consider the full divergence, not just the metric. To be honest, I am not completely sold on the idea of just defining more and more general divergences. It seems to me that one should have a good reason to consider a particular divergence and an application in mind.
For practical purposes, the choice to use the Wasserstein metric vs.
an information geometric divergence really depends on the structure of your problem. Should the distance incorporate the underlying metric of your space or is it just a matter of how far they are from an information theory point of view?


*One can see what happens if you consider a linear combination of the Wasserstein and Fisher-Rao metrics. Two separate groups of people have researched this independently in the last few years, and have some interesting results on the associated metric space. Here are links to two of the papers in this new formulation. (https://arxiv.org/pdf/1508.07941.pdf and https://arxiv.org/pdf/1505.07746.pdf).
These new metrics are useful for studying reaction-diffusion equations, where the information geometry induces reaction while the Wasserstein geometry induces diffusion. It takes some work to make the preceding sentence precise, but it turns out to be a good general principle.
From a meta perspective, information geometry has a much smaller community within the math world than optimal transport does. Part of this is that it is a much younger field, but I also think it currently lacks the sort of punchlines that optimal transport has. In order to motivate a mathematician to consider optimal transport, one can easily point to some of the big theorems and conjectures (e.g. "Nearly round spheres look convex"). I'm not aware of similar results in information geometry. As a final note, classical information geometry is largely based off of affine differential geometry, which is distinct from Riemannian geometry. There seems to have been a small renaissance of affine differential geometry in the past 15 years. One hopes that some of these breakthroughs can be used to develop and stimulate interest in information geometry.
