The term 'Mountains of the Moon', refers to the Rwenzori Mountains of Uganda, which are located northwest of Lake Victoria, 185 miles west of Kampala, between Lake Edward and Lake Albert.
The film is not only based on the historical biographical "Burton and Speke" novel by William Harrison, but as outlined in the opening credits, also on the original journals of the explorers John Hanning Speke and Richard Burton.
Burton's wife is presented as having the heart of an adventurer. In real life, she was tolerant of Burton's more exotic extremes but disapproving of many; specifically his writings on sex (which made his translation of Arabian Nights somewhat of a scandal). At the time of his death, Burton had several thousand pages of notes and contents of a book. These would have been extremely valuable to historians but in the end his wife betrayed him, relating that she had received guidance from God and had therefore burned every single page of those manuscripts.
The spear through Burton's face was an actual event, although the scar as depicted was wrong. The spear split his palate, causing a serious internal fracture of the jaw. It also came close to severing his tongue and it knocked out a half-dozen teeth, as well. The fact that he could survive such an extreme injury was near miraculous.