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:
Polymorphism ... is the ability of a solid material to exist in more than one form or crystal structure. Polymorphism can potentially be found in any crystalline material ...
The allotropy article defines allotropy:
Allotropy or allotropism (coined from Greek "other" + "form") is the property of some chemical elements to exist in two or more different forms ... The term allotropy is used for elements only, not for compounds. The more general term, used for any crystalline material, is polymorphism.
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.
The difference between allotropy and polymorphism is simple, the former talks about elements while the latter talks about anything material, other than an element, a molecule, a compound or even mixture, existing in different crystalline structures due to the conditions the material is placed of temperature and pressure .