What is the difference between a collapsible chair and a fold-up chair?
They are the same piece of furniture.
The kind of metal chair that is used for church halls, classrooms, and other venues is called a folding chair.
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They apparently mean the same thing, but neither would be used in everyday speech in the US. The term used is "folding chair."
The term "collapsible chair" is especially bad-sounding - like the chair will suddenly collapse while you're sitting on it. And the word "fold-up" has bad connotations in the US (meaning go broke or fail). So neither of these terms is used in the US.
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 AlpheccaStars's reply was promoted to an answer.