These aren't synonyms. Random is a property of the world itself; stochastic means "we allow ourselves to think of this process as random because we can't/don't

These aren't synonyms. Random is a property of the world itself; stochastic means "we allow ourselves to think of this process as random because we can't/don't want to deal with it exactly". Following this is the difference between Random Differential Equations and Stochastic Differential Equations.