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Scholastic took care in editing the BSC books for UK readers - in the text they changed the American spellings to British ones and some typically American words to their British-English equivalent. I've listed some of the most common words below, although some may be missed out because i didn't notice them or just because I'm doing this from memory - now in a less confusing table! The easiest way to spot these differences (aside from doing a page-by-page comparison of a US and UK edition, which I have neither the time nor money to do!) is to skip to the handwriting pages in the Specials. It's rather obvious that the offending words were tipp-exed out and then written over - the difference in handwriting stands out. Explanations provided where I saw appropriate.
EDIT: Recently I had the opportunity to read ebook versions of several of the US BSC books. The earlier books are edited a lot more than the later books - in book 1 Sam, Stacey and David Michael play Snakes and Ladders (I had certainly never heard of Candyland until I read of it in an American book), the sweet names are changed to those more typical for the UK, for instance Claudia eats liquorice bootlaces in book 2 (as opposed to 'licorice whips'). The earlier books are also peppered with some very minor edits that I suspect were made to raise the standard of the language used - some typical American English sentence constructions wouldn't be considered acceptable in British English, thus I would hold many of these changes akin to translation. In book 2 Claudia tells Janine that "it's cold outside" because to say that "it's cold out" here doesn't make sense and/or would be considered poor English.
As far as I know there were no title changes in the mystery series, apart from dropping the hyphen in baby-sitters, etc.
The letter books were released under different titles - Chain Letter became Dear Babysitter...Letters From The Babysitters Club and Secret Santa became Merry Christmas, Babysitter...Seasons Greetings From The Babysitters Club. I really don't know why the titles were changed - maybe because chain letters and Secret Santas weren't as common back in the day... - but an interesting thing to note is that the text was left unedited. I suppose that editing the black-and-white handwriting pages in the Specials was easy enough, but the Chain Letter and Secret Santa books were basically picture books and it would've either been too difficult to edit the text (especially those on coloured paper) or too expensive.
The California Diaries have different handwriting typesets inside (for Dawn and Sunny at least - I'm not sure about the rest) - Dawn's in particular is completely different from her BSC handwriting, and in Ducky's first diary when the girls each write in a sentence at the beach, Dawn and Sunny's writing doesn't match the writing in their own diaries...In Sunny's first diary the diagram of how her father spends his time is (I think) kept intact, because the writing is all in capitals and it uses the word 'Mom' where in all other instances it's changed to 'Mum'. I also have a feeling that Maggie's weight was originally in the imperial system in her second California Diary - it's in metric in the UK edition.
Given that the logo is different in the UK version, the text from Kristy's Great Idea is slightly different to the original:
"How about a roof shape with our
initials on it?" said Mary Anne.
Claudia's comment about the alphabet blocks is deleted for obvious reasons.
The only other obvious text difference I can find in the original series at present is in book 17, Mary Anne's Bad-Luck Mystery - when Mary Anne mentions that a black cat crossed her path as she was on her way to school, the sentence is marked with an asterisk and the footnote reads "This is bad luck in America". Which was helpful, because here a black cat crossing one's path is good luck.