Partake is a rather formal verb which most people don't use much. It can mean a) to have a share or part of something along with other people. 'There was food available, but he chose not to partake (He chose not to eat any food) 'Participate' cannot be used in this sense or b) To join with others in doing something 'Let us all partake in the celebrations. (today we would use 'join in' or 'participate' more in this sense) 'Participate' means to be invoved with others doing something. 'Most people joined the game, but a few chose not to participate.' In your examples the more commonly used verb would be to participate 'The young lady participated in the demonstration.' 'The young lady participated in the carnival.'

The correct form of you question is: "What is the difference BETWEEN these words?"

They're the same word.