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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."

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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.

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    Of course, as a mathematician, Edwin, you may find the term "random" a little harder to explain. My son, who is a neuroscientist has researched the quite different ways in which the brain produces a random-number series to the way a computer does it. When he tried to explain the problem to me, the difficulty seemed to boil down to a question about the meaning of "random". What do you actually mean by "random"?
    – WS2
    Aug 28, 2019 at 16:03
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    In everyday life, I assume I use it in a fairly loose way, like most people. If I drop a book on the floor and it lands open, the lower page number will be random. Of course, that's not maths-precisionist, as the bounciness of the protruding covers will mean that very low or (book-)high numbers are less likely than intermediate numbers, and the position of the book when dropped, and height.... 'Random' wrt numbers means having a uniform probability distribution. If I throw the impossible-to-find fair dice 6 million times, I get close to 1 million of each result. / I think subjectiveness ... Aug 28, 2019 at 16:15
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    must defeat any attempt to come up (unaided) with a sequence of random numbers; the brain is probably skewed away from symmetry in subtle ways (doubtless different in each person). If a person consciously tries to compensate internally, they will introduce other biases. One needs the best approximation to a fair dice (using the now commoner singular). Add (Mod 10) the digits of pi to a list of 'random numbers'? // Of course, one could probably convince oneself that one was a perfect random number generator. Aug 28, 2019 at 16:23
  • Yes, the human intersperses a lot of "ers" and "ums" and hesitations, between the numbers. The task was to better understand what was happening, by replicating on a computer what was going on in the brain to produce that result. But my query was whether the fact that the computer, when left alone, seemed to spend less time thinking made its results any more "random".
    – WS2
    Aug 28, 2019 at 17:51
  • The problem with achieving a computer-generated 'random' sequence is getting the truly fair dice into its innards. It's spending time checking for symmetry (hopefully). Aug 28, 2019 at 19:08

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