Even though Shadow Wizard's answer points out that there is a different message shown to the actual question owner, I still think a change in text is warranted. Both the publicly displayed text and the personally displayed text are misleading.
The public text states:
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
This is misleading because if the question has been asked before but the answers do not fully address your question, the appropriate course of action is decidedly not to ask a new question. Any new question will also be closed as a duplicate. The recommended course of action is to try to solicit better answers to the existing question in any number of ways (e.g. comment, bounty, chat, etc.). So the last part of the text telling users to add a new question should be removed, and possible replaced with a directive to solicit better answers to the existing question.
If the "ask a new question" directive is referring to a question that is actually different, then the sentence should not open with "If those answers do not fully address your question" which clearly implies that it is the same question but you are just unsatisfied with the existing answers. It should say "If you have a different question". While this would be more accurate, in truth it would still be unnecessarily confusing. If someone truly has a different question, I doubt they will think that they can't ask it because a different question was closed as a duplicate. The text clearly states that this question has been asked before and already has an answer. Obviously, a different question has not necessarily been already asked and answered. Introducing the suggestion about asking a new question is probably more likely to confuse people into re-asking duplicate questions.
The private text suffers from the same flaw. The text states:
This question was marked as an exact duplicate of an existing question. If this question is different, please edit it to explain how it is different or ask a new question.
The beginning of the second sentence is great; if the question is different then indeed the correct thing to do is to explain how it is different. However, the end of the sentence is misleading. Posting it as a new question is not going to help. If this iteration got closed as a duplicate then the next one will to, unless it explains how it's different. If there is actually an explanation that can show a difference, then that explanation should be edited into the first attempt, and not be posted as a new attempt.
And if the text is referring to a completely different question then it is a non-sequitur. It's not "if this question is different" that you should ask a new question; it's "if you coincidentally happen to have another question" that you should post a new question. And if the person truly has a completely different question, it would seem pretty self-evident that it would not be prevented on account of an earlier question being a duplicate. So once again the text introduces unnecessary confusion.
In light of this, I suggest that the "ask a new question" part of both texts should be removed.