We use "will" at the moment we decide or announce something, and "going to" once it becomes a fixed plan. So if, in your example, the mother offers to help the child, she is deciding at that moment that she will help, and announcing it to the child. For example: "Are you going to lunch now? I'll join you." "I'm going to have lunch with my boss Thursday afternoon." In the first case, the decision is made on the spot; in the second, the appointment is set and probably on a calendar. We also use "will" for predictions ("You will be very happy when I tell you what happened") and to show determination ("I will organize my tax information this weekend, I promise!" or "I won't do it, I don't care what you say!").

It's a good idea to read this discussion of Rilker's (Su.Ki. and others have given some good advice and explanations): http://www.italki.com/discussion/88648 It will also be worth your time to search for "difference will going to" - this question has been asked and answered thousands of times already. :)

However, –Mum, I can't do it on my own. It is too difficult. –I will help you with it. Why I can't use " be going to help " here?

in my opinion if you are planning something in future , you should use to be going to* will; it is future , you can use everywhere.