What is the difference between parentheses and brackets?

So, I’ve been using the word ‘brackets’ to refer to the following: (in brackets) but what exactly are parentheses? Is parentheses just the Latin word for brackets or is there any other difference between both words?

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Parenthesis means “put in beside”. Whereas brackets from Old French braguette means “the opening in the fore part of a pair of breeches”.

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To a common user of the English Language, I’d say, both mean the same. The purpose is to enclose something. Brackets are more popular.
However, a student of science won’t use parentheses when they mean brackets.

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In AmE
Parentheses ()
Brackets. [ ]
Braces or Curly Braces. { }
Braces are used mainly in mathematics I believe.
Parentheses are used for (a parenthetical thought [while brackets enclosed a nested parenthetical]).

When I worked internationally we had
Brackets ( )
Square brackets [ ]
Curly brackets { }
Brackets were used for parenthetical.
Square brackets had a specific editorial use.
I don’t remember a use for curly brackets other than mathematical.

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In mathematics, parenthesis are used to change the default order of operation. Most of my life I thought that mathematics was an international language. Several years ago I realized that there are differences. Order of operation is one of those differences. The use of brackets clarifies the order, making it understood worldwide.

With embedded brackets, I learned to start with parenthesis for the innermost grouping. Then use square brackets. Then curly brackets. Like this: { [ () ] }.

In advanced mathematics, you can run into very large equations with dozens of elements. To work the equation, you don’t look at individual elements. You look for certain patterns and “shapes”. The use of different styles of brackets helps a lot by reducing several elements to a single bracketed element that can be viewed as a whole.

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To me there is a clear difference, which is quite unscientific! Parentheses can be used often between commas, dashes and of course brackets. In spoken speech this can be indicated by the rise and fall of the voice but the part ‘in parentheses’ is essentially an afterthought to what is being said or written.

This is an example -
At the present time - I certainly have never experienced anything like it in my lifetime - the whole world is suffering in equal measure from the current pandemic which is likely to continue for years to come.

Brackets are for me used to give additional information.

This is an example
At the present time (2019 -21) the whole world is suffering in equal measure from the current pandemic which is likely to continue for years to come.

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There is a memory aid that I thought of a number of years ago.

Everyone knows that a sandwich is something you eat. Sandwich can also be a verb - to place closely between two things.

Which vs that:
If you can sandwich words between parenthesis, then use the word which. Otherwise use the word that.

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A good mnemonic!
(But it should be between parentheses)

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I didn’t even know there was a difference in spelling between plural and singular. Maybe because I always think of them coming in pairs, similar to ‘pants’.

edit: I just checked the pronunciation.
pə-rĕn′thĭ-sĭs
pə-rĕn′thĭ-sēz’

I don’t think I have ever heard anyone use the first pronunciation.

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I use it. I teach my students too.
The last syllable is stressed for the plural.

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It’s been a long time since I took an English class, so maybe I don’t remember. I don’t recall the subject of parentheses ever coming up. I think using them would have been very frowned upon. I don’t think I would have ever considered using them.

They came up a lot in math, but that’s a different topic. It was almost always used in the plural. It’s possible that opening or closing parentheses would come up. ( and they would pronounce it pə-rĕn′thĭ-sēz’ )

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I have definitely used the word parenthesis, though only in editing.
“The word before the opening parenthesis is spelled wrong.”

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