Both mean to move or act with haste.
Do the following sound the same?
- I'm in a rush.
- I'm in a hurry.
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While both suggest haste, rush has a stronger implication of carelessness than hurry.
This is not absolute though, to the point where I would likely just sound them out and decide from meter. Still, if I either wanted to explain a mistake or a discomfort - or by the same token, if I wanted to emphasise that the lack of error was in the face of having to do the task quickly - then I would favour rush.