I am not sure I had heard of neem before I spent time in India, but I certainly heard a lot about it while I was living there.
Neem was one of the few things that seemed to cross over from the traditional ways to the consumer culture that was just emerging in the early 2000s: neem was the highlight ingredient in toothpaste, mouthwashes, bottled shampoo and face packs. At the same time, people still spoke about it as one of the old reliables, something to swear by. And what they said was that it was good for anything and everything, a pharmacy that could serve an entire village in one plant.
Neem leaf has been traditionally used to treat everything from leprosy to gingivitis. It is particularly known for its antibacterial and antifungal properties, which makes it useful in treating skin infections and rashes as well as dandruff, itchy scalp etc. (It is also an antiviral.) Being a bitter, it is helpful for digestive problems ranging from lack of appetite and indigestion on one hand as well as diarrhoea on the other.
Another of its special uses is as a form of birth control; it is known as a spermicide but can apparently interfere with conception even when taken internally. Apparently it is also effective as an insect repellant: for people it involves neem oil mixed with witch hazel; for plants, neem oil and liquid soap and water. (Neempedia says it is effective against 500 different species of insects!)
In the Muslim world, it is best known as meswak or miswak, the sticks people use in lieu of cleaning their teeth with a toothbrush, which was a custom of Prophet Muhammad.
A sacred plant, neem is considered a manifestation of Mother Kali and/or Mother Durga. In some representations, neem is a goddess in her own right who holds a neem leaf aloft as an emblem of healing and purification. In addition to healing disease, neem is also used as a smoke or incense to dispel negative energy.