How is naproxen(Aleve) different than ibuprofen (Advil)?
Let's start with how they are the SAME. Both are classified as "NSAIDs", which means Non-Steroid Anti-Inflammatory Drug. So both naproxen and ibuprofen can be used to decrease swelling and inflammation, and both will lower elevated body temperatures (fevers). The product insert explains that naproxen is indicated for the relief of pain and fever, including headaches, toothaches, muscle and back aches, arthritis pains and menstrual cramps. Naproxen lasts longer than ibuprofen, so you only need to take it twice per day (every 12 hours) rather than ibuprofen's recommended 6-8 hours.
A quick perusal of the current medical literature does not reveal any shocking data between ibuprofen and naproxen- both have similar effectiveness in pain relief and in side effect profiles. That being said, in my clinical practice I have traditionally used naproxen as my NSAID of choice for menstrual cramps, especially when there is heavy bleeding along with the cramps. Physicians often use prescription strength* doses of naproxen twice daily for the week before menses, then go to as needed use of naproxen the week of menses. This frequently decreases the amount of menstrual bleeding and improves the cramps associated with periods. Could this be done with ibuprofen too? Yes, but since this is scheduled usage, it's nice to use a medication that is only twice per day versus three times.
All NSAIDs can potentially irritate the stomach lining and/or your kidneys, and they may cause fluid retention (do they make your rings feel tight?) In my clinical experience, though not supported by any recent evidence-based studies that I can find, the shorter acting NSAID ibuprofen often helps more with acute pain (especially injuries) than the longer acting naproxen, but this is really a personal preference.
BOTTOM LINE: Add "Aleve" (naproxen) to your first aid kit, and consider this a first choice for menstrual cramps. Figure out for yourself which type of NSAID (ibuprofen vs naprosyn) seems to work best for your other aches and pains, and talk with your doctor to be sure which one is right for you.