What's the difference between a shamrock and a clover? 

While all shamrocks are clover, not all clovers are shamrocks. Clover is the common name for a number of species in the Trifolium family, with the word trifolium meaning “having three leaves.” Most (but not all) botanists agree that the white clover (Trifolium repens) is the plant referred to as a shamrock.

Irish legend claims that St. Patrick used small clover with three leaves called shamrocks to illustrate Christianity’s holy trinity of Father, Son and Holy Spirit. Four-leaf clovers are the result of rare genetic mutations that occur in three-leaved clovers.
While clovers with four leaves are considered lucky, they are not considered to be shamrocks by those familiar with and passionate about Irish tradition.