通称 refers to a commonly used name or popular name, often an abbreviation or nickname, whereas またの名 simply means 'another name' without implying common usage.
Etymology
A phrase composed of またの (adnominal form of 又, 'again, also') and 名 (name). The exact historical derivation is uncertain, but it has been used as a set expression for 'another name' since at least the Edo period.
