I think I've also seen 一番目 in the context of first line(of text), first sibling, so it's a first of some order/list. 一番 on the otherhand often is used to mean "the most". 一つ目 I've never heard of.. I hope this helps. :)

*If they are out of order. *目 or 番目 was ... Excuse me :(

1 is 一番, and then 一番目 is the first number among numbers. For example, there are numbers from 1 to 5 in order like 1 2 3 4 5. You can tell that 1番目 is １. Then, how about these numbers? like 3 2 5 1 4. If they are out of numbers, you can say 一番目 is ３. 目 or 番目 were created for 歌舞伎 several hundred years ago, and that is the difference between 一つ目 and 1番目. Also, you may use 一つ目 for only a thing, while you can use 1番目 for both a thing and a person.

目means a thing/matter maybe, and it is used when you choose one thing from something a lot. For example, who was your first boyfriend? "一人目の彼氏は誰ですか？" Sorry for my bad explanation:/