What is the difference between Prime Minister and Premier?
A Prime Minister and a Premier are different political leader roles. In Canada the Prime Minister is the leader of the country, to top dog, other than the Governor General, who is really just a figurehead these days. A Premier in Canada is the leader of a province.
There are three levels of government in Canada, federal (country), provincial, and municipal (cities). Each has respective leaders, the federal leader is the Prime Minister, the provincial leader is the Premier, and the municipal leader is the Mayor. There people all work together to run the country, the Premier of course looks to the Prime Minister in regards to his or her province, they know what their province needs and they harass the PM for whatever it is they need.
The United States equivalent for a Prime Minister would be the President and the equivalent for the Premieres would be governors I suppose.
This is only for Canada. But through history there have been Premiers and Prime Ministers for Countrys. Premier is head of state while Prime Minister is head of government. Premier being more ceremonial.