They mean the same thing just used in different context.... "eat your heart out" is used to be funny or to be exceptionally mean. Like, "I'm flying to Hawaii, eat your heart out!" Or, "I'm dating your ex now, and I don't care what you think. You can eat your heart out!"
Anyway, thanks for the reply
In your examples "eat your heart out!" is imperative, so in Russian "Страдай молча!" will work in such situations. It is not a specific set phrase though and many other ways of translation are equally possible.