Carbon atom has 4 electrons in its outer shell. In case of diamond, each outer shell electron of every carbon atom forms a covalent bond in a tetrahedral arrangement, thus forming a rigid structure which means no free electron for charge transport. On the other hand, graphite has hexagonal arrangement in which only three out of four outer shell electrons form covalent bonds, thus leaving one free electron. This delocalised electron can move move freely between the carbon layers of graphite and conduct electricity.

Carbon atom has 4 electrons in its outer shell. In case of diamond, each outer shell electron of every carbon atom forms a covalent bond in a tetrahedral arrangement, thus forming a rigid structure which means no free electron for charge transport. On the other hand, graphite has hexagonal arrangement in which only three out of four outer shell electrons form covalent bonds, thus leaving one free electron. This delocalised electron can move move freely between the carbon layers of graphite and conduct electricity.

Consult Sec. 3.3.2 (p. 73) of [1] (contribution by J-P Issi et al.). Briefly, the interlayer coupling between the graphene sheets gives rise to band overlaps in graphite, turning graphite into a semi-metal.

[1] H Aoki, and MS Dresselhaus, editors, Physics of Graphene (Springer, Heidelberg, 2014).

one direct and short answer is :

Energy States distribution

Due to the Energy States Density Dist. even if the constituent atoms are the same atom , Graphite (and Graphene as well) conducts electricity , but Diamond does not . since it has a different ESD distribution

As Behnam Faridm, 2 days ago.

Also, Carbon is capable of forming many allotropes[1] due to its valency. Well-known forms of carbon include diamond and graphite. Around 500 hypothetical 3-periodic allotropes[1] of carbon are known at the present time according to SACADA database.

1. https://en.wikipedia.org/wiki/Allotropes_of_carbon

This is not rare situation. For example, white phosphorus is dielectric, but red (violet) and black form (allotropes) have significantly other electrical properties.

I appreciate all of you for your valuable answers. Saying this , as I understood from your answers, presence of free ions/polar groups/free elctrons and presence of free spaces b/n atoms/molecules may have a great contribution for electrical conductivity.

i.e free ions/free electrons will transfer electrons while presence of free space will enhance the freedom of electron mobility.