I think that Jmat's answer is right in the majority of cases, but it does vary a little from country to country. In the UK, they are different positions within the British Government. The Chancellor (or Chancellor of the Exchequer) is the head of the Treasury and is responsible for financial matters. This person reports to the Prime Minister.

The are not interchangeable. You must find out what the actual title of the official is and then use that title. You cannot talk about "the chancellor of Canada" or "the prime minister of Mexico." Words like "chancellor" and "prime minister" and "premier" and "president" have precise, technical definitions which you can look up in a dictionary or an encyclopedia, but the definition _might be different in every country._ I do one of two things, depending on how interested I am. I can either shrug and recognize them all as "some very high-ranking official, possibly a head of government, who isn't a king or an emperor or any kind of royalty." Or, I can quickly consult a reference--I really think Wikipedia is very good for this--and spend five minutes learning about the government of country X and what that title means in that country. For example, this is ahdictionary.com's definition for "chancellor:" chan·cel·lor (chănsə-lər, -slər), n 1. Any of various officials of high rank, especially: a. A secretary to a monarch or noble. b. Chiefly British The chief secretary of an embassy. c. The chief minister of state in some European countries. 2. a. The president of certain American universities. b. Chiefly British The honorary or titular head of a university. 3. Law The presiding judge of a court of chancery or equity in some states of the United States and in Great Britain. So if I read "Hitler became chancellor of Germany in 1933," I can say "OK, meaning 1c." And if I read "After the controversial speaker spoke at University of Wisconsin-Milwaukee, Chancellor Mone said 'I strongly condemn the belittling of others and their appearance, and the encouragement of hate and harassment,'" I can say "OK, meaning #2a."

Prime ministers and chancellors have similar roles in that they're the head of government, but what title is used is specific to each country. Australia has a prime minister, but not a chancellor. Germany has a chancellor, but not a prime minister. Is there a practical difference between the two? Not that I'm aware of. A country's choice on whether to use "chancellor" or "prime minister" has more to do with what word the makers of that country's legal system preferred than anything else.

And to add on, we don't have a chancellor or prime minister in the United States government. We do have chancellors that preside over universities and other organizations.