Difference between Brackets and Parentheses
Key Difference: Brackets and parentheses are symbols used for enclosing words or numbers. Brackets generally refer to square or box brackets with symbol [ ], whereas parenthesis or round brackets are depicted by symbol ( ). Both have different usage in mathematics and common English.
Brackets are basically symbols which are used in pairs to represent a group composed of various members. There are various types of brackets like 
Round or parentheses with symbol ()
Square or box brackets with symbol []
Braces or curly brackets with symbol {}
Angle brackets <>
However generally, the term brackets are used for square brackets. In mathematics, it is used for interval notation used for expressing the domain and range of functions. Parenthesis (single one) is used if the point is not being included in the interval, whereas a bracket is used when the point is included.
For example  (5, 6]  This means that 5 is excluded and 6 is included in this interval.
An interval for infinity or negative infinity is always depicted by using parentheses, this is due to the fact that infinity cannot be contained, and therefore expressed by using parentheses in place of a bracket. One can also use reverse brackets to depict parentheses.
Parentheses are generally used in mathematics for grouping expressions which are used for finding out the precedence order in operations. Square brackets in mathematics are generally used for enclosing arguments of functions.
In common written language parentheses are also widely used, especially for keeping apart the subordinate material. For example – the color of the car (stolen by Joe) was red . In this sentence, one can just skip the stuff stolen by Joe, as it does not affect or alter the meaning in its surrounding sentence. It is also used when there is lack of certainty in issues related to gender, subject is singular or plural, etc. On the other hand, square brackets are most commonly used for modifying quoted text by someone who is not the original author.
Like – "She [Lily] was happy to see him [James]"
This can be done for adding missing words, adding some editorial comment, etc. One can also use brackets for including parenthetical material inside parenthetical material like – He was the chairman (a very high position [paid]) for a long time).
Comparison between Brackets and Parentheses:

Brackets 
Parentheses 
Symbol 
[ ] 
() 
Interval notation 
Used for depicting that point is included 
Used for depicted point is not included 
General usage in Mathematics 
For inclosing the arguments of functions Lie the functions Range, Sin, etc. 
For grouping expression and to determine the precedence order of operations 
In common English 
For modifying quoted text by someone who is not the original author 
Especially for keeping apart the subordinate material 
Usage in English 
Comparatively less common 
More common 
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