Difference between die and mould?

Typically, it's an industry-sector distinction: a die is commonly used to form (sheet) metal, through the action of closing on the material to press and form it to shape. A mould usually closes and then has material (often plastic) injected. This is NOT a universal terminology, however. For example, when producing die-cast components in metals such as zinc or aluminium (relatively low temperature melting point), the die closes and the metal is poured/injected.