In the original there were frequent occurrences of the names of various anatomical parts or of various acts that are considered taboo now.
Also there were many explicit descriptions of love scenes, even if a poetic language was used for them.
Of the old translations, only the French translation of Mardrus and the uncensored variant of Burton were somewhat close to the original text. Even the "uncensored" variant of Burton was more distant from the original text than the Mardrus translation.
There exists a much more recent French translation which is said to be more faithful to the original text than any of the previous translations (while the Mardrus translation was not censored, there were many places where it was not a word-by-word translation).
I do not remember precisely, because I have read it a long time ago, but I believe that the new good French translation is that by André Miquel et al.; various editions of that may be found at Amazon. I am not aware of any decent English translation.
When I was very young, I have read a translation into my language of the French translation of Mardrus. The first few volumes matched the French text, so they were quite explicit. Then it seems that someone noticed that and ordered the translators to remove any explicit parts, so the following volumes were obviously censored, lacking many words or entire scenes from the French Mardrus text.
Mardrus got translated into English by Mathers. It's the edition I've read (partially—my copy is in four very fat volumes, so it's a lot of reading). Can confirm that it includes tons of stuff I'd expect to be missing from any kind of Bowdlerized or censored version. Not sure I'd recommend it for a modern reader in English given the other options available, these days.
I've read part of the stories some time ago. What made me the most impression was how much the heroes of the stories spent in thinking and talking about cuckolding. Not sure if this is what the author means though.
I remember reading a story where sailors discovered an island where plant-women were growing. They were friendly and gave fruits growing on them to the sailors, but the few who tried to "embrace" them got absorbed. Yep, there was some vore hentai connoisseurs at the time.
If you read the story "The Porter and the Three Ladies of Baghdad" (I think... this is from memory) the women (the escalating beauty of whom as the porter meets each is incredibly phrased by Burton) eventually are fooling around with the porter and they are pointing to each other's crotches and joking about different names: "thy cleft, thy slit, thy coynte, thy clitoris" and the women suggest "the basil of the bridges" (I confess myself mystified) and "the husked sesame seed" (husked due to female circumcision).
Other translations are muuuuuch more tame. Abridged versions probably skip it entirely. When I first read the unexpurgated Burton my mind was blown on the regular.
Aladdin isn't an example of a story with the prurient elements removed, it's an example of a story that was added much later. Aladdin is not present in Middle Eastern versions of the Arabian Nights at all; it first appears in a book written by a French guy.
I wonder about the "prurient" bits? Anyone perused this version and can share what this is about?
I am quite curious to see if the version I read in my own language also has the same censoring.