Why does graphite conduct electricity whereas diamond does not and they are both allotropes?

Diamond and Graphite have 2 different structures, Diamond has a rigid tetrahedral network whereas Graphite has layers which completely changes the properties of both types of carbon.

Basically graphite has free delocalised electrons which means electricity can be conducted, diamond however does not have any delocalised electrons which means it cannot conduct electricity.

Both diamond and graphite are made up of strong covalent bonds which means they both have high melting points. Graphite is soft though because the layers which it is made up of slide, this makes it soft. Diamond is made up of carbon atoms close to each other and in a 3 layered cube shape which is a lot more rigid and this makes it 'the hardest substance on earth'.

Hope this helps, Obrien9