Can someone please give a clear definition and distinction of these terms, as when a public figure is asked a difficult question and says: "I'll take that on notice" or "I'll take that under advisement."
Let me answer your second question first
take something under advisement reserve judgment while considering something.
It's usually used by a judge when he announces he is going to consider what counsel has said and rule on it later.
As for "I'll take that on notice," I haven't heard that used. Usually the term is to "put someone on notice" meaning to alert someone to a fact and to the fact that you have given them some kind of ultimatum:
The police put Niko Bellic on notice that no further racketeering would be tolerated in Liberty City.
It's a way of telling someone "You have been warned."
In Parliament there is a session called Question time where members can ask ministers questions relating to anything. Generally these questions are without notice. That is to say the recipient has been asked a question without the benefit of receiving it earlier and being able to research the answer first. If the answer is not known the Minster can answer by saying that they are taking the question "on notice". In other words, let me find out and get back to you. The reason they word it this way is because the opposite of a question without notice is a question with notice or "on notice".