A mutation is defined as any change in a DNA sequence away from normal. This implies there is a normal allele that is prevalent in the population and that the mutation changes this to a rare and abnormal variant. In contrast, a polymorphism is a DNA sequence variation that is common in the population
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Mutation results due to DNA sequence changes specifically that happen once an allele is transfered from one generation to another and initiate alterations in the allele status from normal to abnormal. In contrast, gene polymorphism is defined as a variation that occurs in allele in a DNA sequence.
Gene polymorphisms are caused by duplications, deletions, and a mutation of triplication of high quantity of DNA base pairs sequences. In addition, Polymorphisms may occur due to changes inside introns or changes in regions that one or multiple DNA bases that are between genes. If the changes occur in a gene’ coding sequence, then different phenotypes may appear as a result of protein variation that is caused by sequence changes. These changes are located exactly in genes’ coding sequence
Karki et al (BMC Med Genomics - 2015) report these definitions:
mutation: DNA variant present in < 1% of the population;
SNP: DNA variant present ≥ 1% of the population.
Therefore, according to the population selected, a variant could be a SNP or a mutation. And a homozygous SNP could also be disease-causing.
They propose that a mutation should be defined as "result of a recent mutation event which has been detected using as a reference the germline DNA of the same individual". In the paper you can find further discussion about the nomenclature.
Article Defining “mutation” and “polymorphism” in the era of personal genomics