The difference is "into". Cut a piece of paper in half. = the paper is now in two pieces Cut into a piece of paper = start cutting but not finish...the paper is still in one piece but has a cut in it. Cut into your savings = take a piece out of your savings

where's the context? do you mean cut a discussion and cut into a discussion? if some one cut a discussion, the discussion will not go ahead, but if some one cut into a discussion, the discussion will continue.