"No less than nine parts of the margosa nim tree are employed in medicine, and I am not aware of any other plant which produces so many drugs" (Basu & Kirtikar, 1918, Indian Medicinal Plants).
In traditional medicine (especially Siddha and Ayurvedic medicine) Azadirachta indica products are used as anthelmintic, antifungal, antidiabetic, antibacterial, antiviral, contraceptive and sedative. It is particularly prescribed for skin diseases (Basu & Kirtikar, 1918, Indian Medicinal Plants).
The leaves of Azadirachta indica are used to treat skin diseases like eczema, psoriasis or mycosis (the principal active compound is the azadiarachtin).
The leaves (and the fruits) are vermifugal.
An infusion of the leaves is used to treat malaria.
A poultice of the fresh leaves is employed to heal festered wounds (Cowen, 1952, Flowering trees and shrubs in India).
The famous neem oil (also called margosa oil) is obtained from the yellow fruit. This oil is effective in the treatment of leprosy, ulcers and skin diseases.
The neem oil is antiseptic and it is believed to cure rheumatism (external application) (Cowen, 1952, Flowering trees and shrubs in India).
Neem oil is also used for healthy hair, to improve liver function, detoxify the blood, and balance blood sugar levels (https://en.wikipedia.org/wiki/Azadirachta_indica).
People use the twigs as toothbrushes and it is effective in preventing periodontal disease (Orwa C, Mutua A., Kindt R., Jamnadass R., Anthony S., 2009, Agroforestree Database: a tree reference and selection guide).
However, insufficient research has been done to assess the purported benefits of neem. In adults, short-term use of neem is safe, while long-term use may harm the kidneys or liver; in small children, neem oil is toxic and can lead to death. Neem may also cause miscarriages, infertility, and low blood sugar (https://en.wikipedia.org/wiki/Azadirachta_indica).