Since the letters I and J are used interchangeably in Italy, the names Iodice and Jodice are essentially the same. Iodice is the dominant and root name. Those called Jodice were probably originally Iodices who were given the name Jodice by some civil or church officials.
I have been researching the Iodice family for many years. I even recovered a genealogy of the family in the national libarary in Naples. The history of the family name goes something like this. The first Iodices came from Rome, but were shipwrecked in Amalfi in the early middle ages. There they founded the cities of Amalfi, Ravello, and Scala. They became merchants and soon expanded all over Italy, especially within the Kingdom of Naples. Their largest concentration is in Naples, Bari, and Caserta.
The name Iodice means Judge. It has been modifed to fit local naming practices, often appearing as the tuscanized "del Giudice" or the apulian Lo Iodice.
My branch of the Iodices come from area around Capua within the province of Caserta.