Shire is from an old Anglo Saxon word and it survived the Norman invasion . So its an administrative subdivision for a region as is county.

In feudal times it corresponded to an Earldom. And in mainland Europe where the Normans came from the corresponding title to an Earl is a Count. So County is from the French and the Normans were from Normandy in France .

So it's pre norman in England and the Anglo Saxon term survived. There are some areas in England not called Shire , like Essex and Co Durham, but they are few.

In Ireland , English was not spoken and the Norman use prevailed.

Sheriff is also Anglo Saxon from "shire reeve" for a magistrate so it hasnt disappeared totally from use.

And in America, counties are mostly named after a person. So take Yorkshire, it's named after the largest town . County Cork is too but there was an Earl of Cork so it's geographical. Cook County, Illinois is named for Daniel Pope Cook.

Hence the difference.