Why is мать considered an error, when the word is 'mother'? As far as I know, мать means 'mother', while мама equals 'mom'.
you are correct.
Generally, it is as you said. However, when you address a child, it is uncommon to refer to their mother as "мать", "мама" sounds better in this context. Both should be accepted, though, I'd report it.
Мать is, unlike мама, a formal word, so with a pronoun ты you'd use мама, while for вы the varians мать sounds better, because then the formality level is balanced. :)
Ok - I've been taught that мама is very informal, almost children's language. I think that both мама and мать should be possible answers :)