What's the difference between stochastic and random?

A variable is random. A process is stochastic. Apart from this difference, the two words are synonyms.

There is an anecdote about the notion of stochastic processes. They say that when Khinchin wrote his seminal paper "Correlation theory for stationary stochastic processes", this did not go well with Soviet authorities. The reason is that the notion of random process used by Khinchin contradicted dialectical materialism. In diamat, all processes in nature are characterized by deterministic development, transformation etc, so the phrase "random process" itself sounded paradoxically. Therefore, Khinchin had to change the name. After some search, he came up with the term stochastic, from στοχαστικὴ τέχνη, the Greek title of Ars conjectandi. Being popularized later by Feller and Doob, this became a standard notion in English and German literature.

Funny enough, in Russian literature the term "stochastic processes" did not live for long. The 1956 Russian translation of Doob's monograph by this name was already entitled Вероятностные процессы (probabilistic processes), and now the standard name is случайный процесс (random process).

Neither word by itself has a commonly accepted formal definition in mathematics, so one cannot really ask about "the difference" between them.

They are used in phrases such as "random variable," "random walk," "stochastic process," "stochastically complete," etc, which have accepted definitions of their own. In all cases both words tend to refer to an element of chance or unpredictability. But they are generally not interchangeable; if you talk about a "stochastic walk" people will be confused.

Random process and stochastic process are completely interchangeable (at least in many books on the subject). Although once upon a time "stochastic" (process) generally meant things that are randomly changing over time (and not space). See relevant citations:

https://en.wikipedia.org/wiki/Stochastic_process#Terminology

In English the word "stochastic" is technical and most English speakers wouldn't know it, whereas, from my experience, many German speakers are more familiar with the word "Stochastik", which they use in school when studying probability.

The word "stochastic" ultimately comes from Greek, but it first gained its current sense, meaning "random", in German starting in 1917, when Sergei Bortkiewicz used it. Bortkiewicz had drawn inspiration from the book on probability by Jakob (or Jacques) Bernoulli, Ars Conjectandi. In the book, published 1713, Bernoulli used the phrase "Ars Conjectandi sive Stochastice", meaning the art of conjecturing. After being used in German, the word "stochastic" was later adopted into English by Joseph Doob in the 1930s, who cited a paper on stochastic procsses written in German by Aleksandr Khinchin.

https://en.wikipedia.org/wiki/Stochastic_process#Etymology

The use of the term "random process" pre-dates that of "stochastic process" by four or so decades.

Although in English the word "random" does come from French, I strongly doubt it ever meant random in French. In fact, it originally was a noun in English meaning something like "great speed". It's related to the French word "randonée" (meaning hike or trek), which is still used today. To describe a random variable, French uses the word "aléatoire", stemming from the Latin word for dice (which features in a famous quote "Alea iacta est." by Julius Caesar). The English equivalent "aleatory" is not commonly used (at least in my random circles).

Stochastic comes from Ancient Greek whereas random is an old French word. (fun fact: random has totally disappeared in modern French and was replaced by aléatoire which comes from... Latin)

Otherwise there is no difference between them in the realm of Probability Theory.