Well that's an interesting question that sounds very familiar to me, since I believe I have asked it before. The answer was something to the effect that they are in fact different in that one is direct and the other is private. So apparently they aren't the same. Hope you get a better result.
While they both send messages to another Insider, the difference lies in the way in which the recipient sees the message.
Direct and private messages may at first appear to be the same thing, but they are two different ways to send messages. Check out the FAQ above to try the differences first hand. Direct messages appear on the recipient's Communications page along with other mentions, shares or replies, while private messages are only visible from your profile page.