Can Differential Privacy be used to show that two distributions are indistinguishable?
These sorts of arguments come up in the theory of randomness extractors where we argue that a random function from a certain family when applied to inputs with a certain minimum entropy ($H_\infty$, the -ve log-maximum probability of an outcome in the Renyi notation) produce outputs that are indistinguishable from uniform up to some $\epsilon$ bound. The differential privacy of the uniform and extractor distributions can be bounded in terms of input size, output size and $H_\infty$ (see for example the Kamp and Zuckerman theorem of the article).
The min-entropy should be the choice of entropy function when using differential privacy as it bounds the worst case scenario of concentration of inputs.