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.