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.