Hi! 「死」 to die (the act of dying). 「亡」 passing away, disappearing or leaving. Examples: 彼は癌で死んだ。 He died of cancer. 老人は今朝亡くなった。 The old man passed away this morning.

Personally, I think 亡 is a little bit formal while 死 is common. What's more, the subject of 亡 tends to be human or human-related things, such as a country and there is some usages like 亡國(the demise of the country), 亡靈(the dead's spirit), etc. 死 can be used for almost all things, like the fauna and flora, individuals and so on. 兔死狐悲; 君要臣死,臣不得不死;一潭死水(a stagnant pond), etc. are some examples about it. In parenthesis, this is Chinese usage and I am not sure whether it is the case in Japanese.