Consider the following. You're looking at a thread on your beloved MSE where somebody has posed a problem they're looking for help on, and you realize you have an answer. You scroll down, and see another answer which does what you'd wanted to, but find that it wasn't written very clearly, or the person made his answer more complicated than was necessary. You want to answer the question, but don't want to be accused of posting the same answer, and don't want to appear rude by sounding like you're trying to one-up someone. What is the proper etiquette?
If your presentation is significantly different*, and you really believe it makes for a better answer, then in my opinion posting a new answer is the thing to do. I've seen it done many times. If it's just a few minor changes that you can apply to the already-existing question, then you can edit it. This addresses "posting the same answer". You can even explain at the bottom of your new answer that your approach is equivalent to the other answer, but that you think your presentation is clearer.
As for rudeness... I don't believe it's rude at all. If someone is offended because they posted an answer on math.SE and you post a better answer, they're the one with a problem, not you. Considerations about ego shouldn't be relevant when deciding to post an answer or what to write in it; if your answer is going to help the OP and future readers, by all means, post it.
* The murky point here is what does "significantly different" means. As a rule of thumb, if you can imagine taking the other answer and adding and removing sentences here and there to get your new answer, it's not different enough. If the answer is completely reorganized and you essentially have to start from scratch, it should be okay.
To repeat Carl's comment, one may post an answer and check the Community Wiki box on the lower right (while still editing); the last time I did this, I also removed a comment on the other answer suggesting changes. The most recent time I did this, I felt the other guy's answer was likely to go over the head of the person asking, while a more accessible demonstration was not difficult to compose. I'm afraid I can't find the specific item, I think it was in May.
This is a compromise, of course. The other guy might dislike the whole thing. However, it is less confrontational than comments saying another answer is not good. It also physically separates the two. Then the CW status says this really is derivative of another answer; third and fourth parties are unlikely to find this annoying.