Their differences are due to the fact that graphite contains impurities that weaken its structure, while diamond is always a pure form of carbon. 

Their differences are due to the environments they form in: graphite forms from organic material, whereas diamond forms from volcanic gases, resulting in different strengths. 

Their differences are due to how the carbon atom bonds and atomic arrangement;  diamond’s carbon atoms have all covalent bonds while graphite's carbon atoms form strong sheets which are held together by weak intermolecular forces. 

Silicon-carbonate octahedron

Silicon-oxygen tetrahedron

Olivine silicate chain 

Calcium-carbonate tetrahedron

amphibole

feldspar

pyroxene

olivine

To be a mineral, it does not need to have a defined chemical composition. 

To be a mineral, it must inorganic.

To be a mineral, it must be a solid when at room temperature.

To be a mineral, it needs to have a regular crystal structure. 

To be a mineral, it has to be naturally occurring.

sulfates

halides

sulfides

halogens

silicates

Si, Al, Fe, O, Mg, Ca, Na, K

Fe, O, Mg, Si, Al, Ca, Na, K

O, Si, Al, Fe, Ca, Na, K, Mg