...but the way they do so differs between/among cultures.
Both versions are valid.
If the same question had been asked back before the 1950s, a grammarian would probably have insisted on using among because there are more than two tea-drinking cultures.
In the debate between vs among, Merriam-Webster says
We use between when we want to express a relation to things and have them considered as individual and usually equal entities: between the devil and the deep blue sea; the restaurant between my house and my work; a treaty between nations. This connotation stems from an earlier use of between that referred to a point between two places, or travel between two specific points. Among, on the other hand is the best word to use when referring to things collectively and imprecisely: for this reason, among many others; no honor among thieves.
In the OP's example, one could argue that among is preferable because the names of the tea-drinking cultures are not specified, unlike in this following example:
The differences between English, Chinese, and Arabic are significant
whereas no university is mentioned except in the subordinate clause
Among all the universities in the UK, Cambridge and Oxford are the oldest and most prestigious.
Although the names of the countries, in Merriam-Webster's example, are not mentioned–“a treaty between nations”– it is appropriate because of each nation has a one-to-one relationship regardless of the unspecified number.