There were many theocratic states in Germany, taking OP's definition of a theocracy as a state in which the civil power is vested in an ecclesiastical official. One example was the Hochstift of Cologne, an area not identical with the Archdiocese of Cologne. This area was ruled by the Prince-Archbishop.
The position of Archbishop of Cologne was, for centuries, filled by a member of the Bavarian ruling dynasty, usually a younger son of one ruler and brother of the next. Joseph Clemens was appointed archbishop at the age of 16 in 1688.
Since the archbishop held the civil power, it could be seen as a theocracy. In practice, far from theocratising the state, it secularised the Church.
Even in the case of the Papal States the ruler, the Pope, was usually drawn from the ruling families and was expected to appoint his nephews as cardinals. It is questionable whether at conclaves the good of the Catholic Church was uppermost in all cardinals' minds. With whom would a papabile side was always an issue.
Luther's objection to the sale of indulgences was widely approved in Germany in part because it could be seen as the abuse by the rulers of Central Italy of the fact that they were, ex officio, popes. It is a perspective which is easy for moderns to overlook.
Pope Sixtus V in the sixteenth century devoted a lot of time and energy, and his subjects money raised through very heavy taxation, to rebuilding and beautifying Rome. He brought in vigorous methods to reduce crime, executing around 5,000 brigands. He tried to stamp out corruption.
Church of England Bishop Lee of Lichfield was credited with hanging 5,000 Welshmen as well as "some of the best blood in Shropshire".
Until 1836 he Protestant (i.e. Church of England) Bishop of Durham was known as Pronce-Bishop within the county of Durham, where he had the same powers as the monarch did in the rest of England. Shute Barrington was bishop from 1791 to 1826. He was responsible for many public works. When the miners in pits owned by the Diocese went on strike he sent in troops against them and came down very harshly on poachers even in a time of hunger.
It would be hard to say there was any significant difference between Protestant and Catholic theocracies though there were enormous differences in conditions at differing times and places. It is perhaps more a question if there was any difference between a prince-bishop and any other prince.
Mount Athos is an autonomous peninsula under the direct authority of the Ecumenical Patriarch of Constantinople. However it contains only monasteries and no civilian population so perhaps is not truly theocratic government in the general sense.