In American English, round brackets "()" are called parentheses and square brackets are thought of as something other than parentheses. In British English, parentheses refer to square brackets "[]." While the symbols differ slightly depending on language context, the meaning is the same -- parenthetical information is information that is inserted into text. It's generally a supplementary detail, supporting the main sentence. Each form of bracket denotes parenthetical information of some sort.
In American English, square brackets are sometimes used interchangeably with parentheses. Though the meaning is probably still understood, it is a misuse of the the square bracket. While they still denote parenthetical information, square brackets tell the reader who is inserting the information. This can be an important distinction, particularly in journalism, where the author may be quoting someone. Square brackets attribute their content to the author, not the person being quoted, whereas quotes don't indicate that the author is augmenting the words of somebody else.
Square brackets are used in written expression, but they are have specific meaning in math and science. Specifically, in a mathematical expression, an expression encompassed by square brackets is to be considered as a unit of expression. They expand upon the order of operations which tells the reader what parts of the expression to evaluate first. Brackets tell the reader to evaluate the contents of the brackets before completing the evaluation of the rest of the expression.
Curly brackets are commonly used in computer programs. In application programming and markup languages, the type of brackets used is critical. Different languages interpret brackets differently. So, information set off with parenthesis may have a different meaning than the same information set off with curly brackets. Programming also uses lots of nested brackets (brackets and bracket sets within brackets) which rely on the sequence of opening and closing brackets to derive contextual meaning.