Kernel mean embeddings and maximum mean discrepancies (MMD) associated with positive definite kernels are important tools in machine learning that allow to compare probability measures and sample distributions. We provide a full characterization of translation invariant MMDs on $\mathbb{R}^d$ that are parametrized by a spectral measure and a semi-definite positive symmetric matrix. Furthermore, we investigate the connections between translation invariant MMDs and Wasserstein distances on $\mathbb{R}^d$. We show in particular that convergence with respect to the MMD associated with the Energy Kernel of order $\alpha\in(0,1)$ implies convergence with respect to the Wasserstein distance of order $\beta<\alpha$. We also provide examples of kernels metrizing the Wasserstein space of order $\alpha\geq 1$. A short numerical experiment illustrates our findings in the framework of the one-sample-test.