"Here is mom."
Вот is used when you're pointing at something or showing someone what is directly in their faces. Usually, it is accompanied by finger-pointing and/or it is easily accessible with your eyes. It's kind of emphatic, which is why it comes first in a sentence ("Here it is!" "Here's your taxi! Right on time!").
Здесь, is kind of the same thing, but you can use it when something is not as easily accessible, like for finding something/one ("Where's Johnny!" "Here I am!") or for any instance of "here" that doesn't involve finger-pointing something out to someone ("Here lies a book." "I used to live here.")
I have the same question... I think it is related to the order of the words. I think that "Mum is here" translates to "мам здесь" meanwhile "here is mum" translates to "Вот мама". I think this because of a previous answer in the comments to a very similar question: "Mum is there" vS. "There is mum". :) Let´s wait until anyone more experienced gives us a certain answer ^_^
Вот usually use with point finger
If you can see the person who asked, and he can see mom, and you can point the finger at her, then вот мама.
If you anwer by phone or shout to person who is in the another room and cannot see you and her, then мама здесь because he is there and you and she are here.
I just want to report a bug: I picked a word from the balloons, but I erased it and it has left a blank space as if the word was still there; I couldn't take it away of there and that led Duo to mark as an incorrect answer, even though I entered the correct answer. I'm on Android.