The difference between mirror and common carp is both genetic and visual - biologically they are similar. The mirror carp was the first mutation of common carp, owing to two alternative genes, the S allele and the N allele. The genetic term for a mirror carp is "ssnn" (all minor). Common carp have an even, regular scale pattern, whereas mirrors have irregular and patchy scaling, making many fish unique and possible to identify by sight, leading to most carp in the UK over 40lb being nicknamed. This lack of scales is widely believed to have been bred in by monks in order to make the fish easier to prepare for the table. They can grow in excess of 60lb - the last few British record fish have all been mirror carp.
Contrary to popular belief, Leather carp are not Mirror carp without scales; there is a distinct genetic difference. Leather carp are permitted a few scales; however, the dorsal row of scales is either absent or incomplete. Leathers also have reduced numbers of red blood cells, slowing growth rates.
http://en.wikipedia.org/wiki/Mirror_carp