Chemistry
Diamonds are made of pure carbon, which makes its chemical formula C. Tanzanite is a variety of zoisite [Ca2Al3Si3O12(OH)] that has vanadium traces.
Color
Diamonds come in a range of colors other than clear including shades of blue, orange, brown and yellow all from impurities, and pink (from a slight natural difference on the molecular level). Diamonds can be heat treated to improve coloration.
Tanzanite is blue to blue-purple. Other colors, colored by magnesium or chromium, are called fancy tanzanites. When heated tanzanite turns deep blue, at times with traces of purple.
Hardness
Tanzanite is softer than diamond. O the Mohs scale of of hardness (a relative scale of 1 to 10) tanzanite rates 6.5 to 7 while diamond rates 10.
Mineral Structure
The mineral structure of tanzanite is orthorhombic while the mineral structure of diamond is cubic.
Sources
Tanzanite is only found in one place, in northern Tanzania mainly in the hills of Merelani, which is located in the Umba Valley in the Arusha region, west of Kilimanjaro.
Diamonds on the other hand are mined in Russia, Canada, India, Borneo, Australia and more than any place else, Africa.
Considerations
Tanzanite should not touch acidic substances or be placed in an ultrasonic bath or steam bath. Diamonds should not be cleaned using an ultrasonic bath due to potentially increasing flaws.