What's the difference between "stochastic" and "random"?

Is one just used to sound fancy?

Webster defines the former as the latter:

Definition of stochastic
1 : RANDOM
specifically : involving a random variable
a stochastic process
2 : involving chance or probability : PROBABILISTIC
a stochastic model of radiation-induced mutation

It seems given the etymology of stochastic ("to aim at"), it would mean something a little less than "random."

Solution 1:

Wikipedia certainly emphasises the synonymity:

Stochastic refers to a randomly determined process. The word first appeared in English to describe a mathematical object called a stochastic process, but now in mathematics the terms stochastic process and random process are considered interchangeable. The word, with its current definition meaning random, came from German ...

but adds the domain caveat:

The term stochastic is used in many different fields, particularly where stochastic or random processes are used to represent systems or phenomena that seem to change in a random way. The term is used in the physical sciences such as biology, chemistry, ecology, neuroscience, and physics as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, (including the field of artificial intelligence), cryptography and telecommunications. It is also used in finance, due to seemingly random changes in financial markets as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.

One wouldn't normally speak of 'stochastic acts of violence' or 'A stochastic collection of tips on kitchens that may come in handy', or find a book titled 'Stochastic Thoughts on Life' by Raymond Lammie for instance.