The semicolon is part of the division of the expression of thought in supersets and subsets of structural affinity: comma, semicolon, period, new paragraph. (Which also means that the semicolon has a necessary role in general: whenever the structure is best defined also using that level of affinity.)
The dash, the colon and the brackets are instead related to the modality of relation: the colon to express dependency, the dash to provide detail, the brackets to insert a note parallel to the main flow.
(Note: there is also a sub-function of the semicolon to separate list items. I did not use it in the paragraph just above as I intended to exploit the "nuclear" aspect of comma-separated list items, but that is a rhetorical option and the semicolon would have been necessary for more complex items.)