I can't answer for every moderator, and the closest thing I found to an official policy is this question on meta.SE, but I can tell you my perspective.
In the case you describe, where a user has written a new answer that isn't the exact same text as an old one but has the same information, typically I'll leave a comment telling the new user that their answer doesn't add anything new, and hinting that it would be better to spend their time on unanswered questions. If there's any possibility the new answer might be useful as a complement to the old one (e.g. it explains things differently, some users might find the new answer clearer, it's updated for a newer version), then I'll leave the answer in place. If it adds nothing at all, I'll usually delete the new answer: not to punish the answerer, but just to keep the site tidy, and to reinforce the message that we're not interested in duplicate content.
As a non-moderator, you can add such a comment yourself, but please make sure to keep it encouraging. The new user is probably used to a forum site where you have to repeat replies to show that you agree with them, instead of being able to upvote. They spent their time writing the duplicate answer because they wanted to help, so complaining at them can feel very negative even if it doesn't seem that way to you. Make sure to say that there's no need to write a new answer, rather than that it's "wrong". Tell them that answering unanswered questions is more helpful. Welcome them to the site and thank them for contributing.
If that all sounds like a lot of work, there's no need for you to do anything. Having the duplicate answer doesn't really harm the site, it just makes it a little harder for the best answer to "rise to the top" with upvotes. If you want to add a custom flag to the answer to let the moderators know, feel free, but the outcome will probably depend on which moderator happens to look at the flag. :-)
There's another case, which is when a user posts the same exact text as another answer, either by the same or a different user, on the same or a different question. Please always flag these. We'll typically delete the duplicate and tell the poster why, but sometimes the situation is more complex and requires more effort. Sometimes it's a mistake, and the duplicate should have been an edit; sometimes it's plagiarism; sometimes it's an honest attempt to help, but it lets us know that there were duplicate questions that should have been closed. The system does quite a good job of detecting them, but some duplicates slip through. Typically I'll always mark those flags as helpful, even if I end up taking no action.
