I am trying to translate the phrase: 「先住民」又は「先住民族」 into English in referrence to international laws and treaties concerning indigenous people, and I am not certain what the difference is between the two, or at least how they would be translated differently in English.
I think the problem with "aboriginal race" is that "race" is normally understood to refer to a group defined in terms of a shared genetic heritage and is usually rendered in Japanese as 人種 (じんしゅ) , while 民族 (みんぞく) corresponds more closely to English "ethnic group", that is, a group defined in terms of language, culture, etc.
The English terms that come to mind (my mind, anyway) for dealing with this translation problem are:
Indigenous peoples, native peoples, aboriginal peoples, autochthonous peoples, indigenes.
There are problems with the word "native" - certainly with the use of "natives" as a collective term for indigenous people - and with "aborigines", because it is nowadays pretty much restricted to the indigenous peoples of Australia (although in the 19th century it was commonly used of Native American peoples). I think "indigene" is an unusual word. "Autochthonous" is probably uncommon, although I use it myself in situations like this. I would, if I had to, probably go with "'indigenous peoples' or 'autochthonous peoples'".
I don't know the context, but it looks to me as though the writer is offering the two terms as ways of referring to the same groups. If that's so, perhaps you could be bold and simply translate with a single term, "indigenous peoples". However, since you're dealing with laws and treaties it is probably necessary to translate in such a way as to show that the Japanese uses two terms. One possibility might be "'indigenous peoples' (senjumin) or 'indigenous ethnic groups' (senjuminzoku)'". This would indicate to the reader that the original Japanese is using two very similar words.