Is diamond an allotrope or polymorph of carbon? When is something considered an allotrope vs a polymorph?
Allotropy is a special case of polymorphism. Polymorphism is the phenomenon of a substance exhibiting different crystal structures. Allotropism is the same phenomenon limited to the subset of all substances that contains only the chemical elements. According to the Wikipedia article on polymorphism:
The allotropy article defines allotropy:
Thus, diamond and graphite represent two different allotropes of carbon, which also makes them different polymorphs of carbon.
Water is well-known for having multiple solid phases. These phases are polymorphs, but, because water is not an element, they are not allotropes.
Diamond is an Allotrope of Carbon.
As cleared by @Ben Norris,
Allotropy is property of any element to exist in two or more different forms.
Whereas the term Polymorphism meant the ability of a solid material to exist in more than one form or crystal structure.
In diamond, each carbon (an element) is bonded to four Carbon atoms forming a rigid 3-dimensional structure.