The Babel documentation lists both options but I did not find where it describes the difference. I found one Brazilian website saying "Use a opção brazil para escrever no nosso português" but it never mentioned the other option.

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    There is no difference, they are synonyms, one in Portuguese and the other in English. – gernot Sep 24 '16 at 13:18
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    @gernot You are probably right about the motivation. But in fact "Brazil" and "Brazilian" are both English. In Brazilian Portuguese they are Brasil and brasileiro. – Colin McLarty Sep 24 '16 at 16:50

Reading the package documentation of babel-portuges:

Now we have to decide whether this language definition file was loaded for Portuguese or Brasilian use. This can be done by checking the contents of \CurrentOption. When it doesn't contain either portuges or portuguese we make \bbl@tempb empty.

I would say that brazil and brazilian are equivalent.

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