Bracket vs brace

"Brackets" is used interchangeably for various symbols:

• angle brackets for <>
• round brackets for ()
• curly brackets for {}
• square brackets for []
• and more, some of which are only used in certain regions or cultures

The more specific "braces" and "parentheses" are almost never used for other than {} and (), respectively. Mathematics has several more (subtle) variations in addition and is formalized more than other fields.

I prefer "braces" for {}, "parens" (informally or in the context of programming) or "parentheses" for (), and plain "brackets" for [], though "square brackets" is sometimes required to be explicit about the latter. This gives you a one-word name for any of those common symbols. (And don't get me started on the problems of angle brackets for grouping... :P)

To provide a non-text counterpoint, "bracketing" in photography is taking multiple exposures around a given value (which value—white balance, ISO, aperture, etc.—depends on what you're targeting). This shows the use of "bracket" to mean "surrounding," which lines up nicely with the text symbols above.

I am not sure about other contexts, but in texts about programming you will often find these terms used as follows:

braces: {} parentheses: () brackets: [] angle brackets: <>

Square brackets [] are often simply called brackets, while curly brackets {} are often called braces (in the US, at least). Someone using the term braces for square brackets is making a mistake, because most people will assume that person means curly brackets. So, these terms are not interchangeable, although braces are a type of bracket.