English has more than one use for the word "first". For example:
- First, we asked our parents. (=at the beginning)
That's the kind of expressions you would only use 最初 for, which is pretty obvious from the kanji it uses. As far as I know 一番 and its variants mean "first" as a number and also "the best".
最初 is also used as "at the beginning/ the one in the beginning/ first time" and so on. A few good examples from Tatoeba:
彼女は最初その馬が気に入らなかった。~ She did not like that horse at first
私が最初に行きます。~I came first (I was the one who came first)
最初の信号を左へ曲がって下さい。~At the first traffic light turn left.