This question seems to be tricky even for native speakers. I have done a quick search on Google and gotten mixed results. However, the best result I have gotten so far is from iTalki.com:
They are pretty much synonyms. I see that m-w.com lists each as a
synonym for the other.TO ME, "excerpt" SUGGESTS a long passage that is intended to be read,
possibly for enjoyment, as a coherent whole. An example would be an
entire book chapter–for example, a whole chapter of a new book
published in a magazine.TO ME, an "extract" is usually shorter than an "excerpt" but longer
than a "quotation."
The sense I am interested in is when used to deal with text whether it is a book, essay, transcript, poem etc.
Some criteria you could think about giving the best answer is:
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Length (which one is longer in general)
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Collocations (I really want to know if there is, and hopefully get some examples)
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Register (academic, governmental, legal, entertainment and spoken English)
Best Answer
As nouns, excerpt and extract are synonymous, i.e. a small piece of a longer text.
In respect to texts, I suspect excerpt would be the more common of the two, mainly because the verb to extract has a much wider range of meanings. One can extract ore, or extract the truth, or extract a tooth; but text is the only thing one can excerpt.