Your translations in English don't make much sense and I probably don't know enough Spanish to understand what you're trying to say. I think perhaps you want to say something like "Do not be in a hurry to get older?" In other words, "Try to stay young for longer" or "Enjoy the present."
Going back to your original question, there is almost no difference between rush and hurry, especially in your example. Rush is usually a verb and hurry is a usually noun. However you can also use rush as a noun and hurry as a verb.
- Don't hurry to get older. (this sounds ok) (hurry = verb)
- Don't rush to get older. (but this sounds better) (rush = verb)
However these are both equally good:
- Don't be in a rush to get older. (rush = noun)
- Don't be in a hurry to get older. (hurry = noun)
The only small difference between these two words that I can think of, is that hurry implies a direction or destination, while rush does not.
Commuters were all in a rush. (no direction)
Commuters were in a hurry to get to the bus. (direction = bus)