Fief is a linguistic variant of feudum.
The connection is a strong one. Whatever the internal linguistic derivation, Tolkien's use of the word fief in the text of LOTR is a telling one. Shire is a more loose term probably derived in most cases from the territory controlled by a tribe or dynastic leader in the early Anglo-Saxon period. Later used as an administrative unit. The Sherrif or Shire-reeve was the crown's administrator in the shire, but he had to work within the feudal society which existed there.