The difference is in the structure or how the Carbon electrons are bonded together.
graphite structure
diamond structure
This makes Carbon really unique and interesting.
Chemical Name: Carbon
Chemical Symbol: C
Atomic Number: 6
Atomic Mass: 12
Carbon has 4 electron in it's out most shell (electronic configuration 1s2 2s2 2p2), it needs 4 more electrons to be stable (read my post on element stability).
The ability of Carbon to bond with itself and to create single, double and triple bonds is another of it's unique characteristic.
To explain all this is too much for a post, so I plan to make separate posts (I hope so) on it. The study of Carbon itself and its compounds is called Organic Chemistry.