First of all, there's an official blog post on handling duplicate questions. It differentiates between three types of dupes:
Cut-and-paste duplicate questions. These questions are the very definition of exact duplicates; they are typically from users who willfully take the very same question and post it again. Either they’re not satisfied with the speed of answer, or they just don’t know what they’re doing. We rely on [site] users to vote down these “questions” and flag them for moderator attention. These sorts of duplicates are typically deleted as soon as we see them, as they’re borderline abuse of the system.
Accidental duplicates. These questions aren’t copy and paste, but they cover the exact same ground as an earlier Stack Overflow question. The overlap is not ambiguous; the question uses the same words and asks the same fundamental question, with no variation at all. This is a failing on several levels; of the asker to do proper diligence before asking, of our internal ask page title search, and possibly of Google search as well. We rely on [site] users to link these questions together by closing them as “exact duplicate” and posting the URL (as a comment, or edit) to the question this is a duplicate of. These sometimes have multiple good answers attached to each question. We will use our new moderator question merge function to merge them together without losing any answers or comments.
Borderline duplicates. These questions are ambiguous; they’re in the same ballpark as a previous question, but have subtle differences that may make them legitimately standalone questions. These are subject to interpretation. We rely on [site] users to tag these questions appropriately so they naturally “group” with the questions they’re related to. The more tags the questions have in common, the more likely they are to show up together on the related questions sidebar. You can also edit in links to the possibly duplicated posts, if appropriate, but be sure to make the tags match so the system can figure out the relationship without as much manual effort. There’s often benefit to having multiple subtle variants of a question around, as people tend to ask and search using completely different words, and the better our coverage, the better odds our fellow programmers can find the answer they’re looking for.
(Emphasis slightly adjusted.)
Now, we don't have that many borderline duplicates, and I don't think I have ever seen a cut-and-paste duplicate at all. Most of our dupes so far are of the accidental variety.
That's not surprising, given that the site search still doesn't work for a whole lot of stop words. Consequently, I would say that our very own most important rule is not to blame the OP for not being able to search. From all I can tell, this is not happening anyway, but I think I should mention that for future reference. Instead, we should simply point the OP to the original question in the comments. Note that such a pointer is automatically generated by the system once a question is voted to be closed as a dupe.
As to deleting, I would say that while accidental duplicates should be closed, they shouldn't necessarily be deleted. Quoting Jeff Atwood,
As I’ve mentioned probably ad nauseam at this point, people have the uncanniest knack of asking multiple, nearly identical questions with almost zero words in common.
And that's not necessarily a bad thing. By having closed-but-not-deleted duplicates, we have several completely different wordings pointing to the same question. For one, that increases our visibility in Google search results. Most importantly though, it helps prevent more dupes from being posted in the future. Whenever you try to post a question, there's the Related Questions box that hints at what your question might be a dupe of. The more different wordings of the same question it has to compare your wording to, the better.
Lastly, as Robert Cartaino says over at Gaming,
Closed does not (necessarily) mean "does not belong on the site."
Closed means "no longer accepting answers." [...] In terms of "does not belong on the site," That's what
delete is for.
So, just because someone asks the same question using a different wording, doesn't mean it doesn't belong. After all, it is still on-topic; thus, it doesn't have to be deleted. It's just that it has been answered already; thus, it should be closed.