В чем разница между remain и stay ?Пожалуйста, приведите примеры.

They mean the same thing. They are synonyms of one another. However, there can be a difference in use. I would use "remain" in more formal speech and "stay" in any type of speech, but others may not.

For example, if I were writing to a superior about an issue I would say "I would like to remain in San Francisco, rather than transfer to Los Angeles." just because it sounds more proper.

Although, there is nothing wrong with saying "I would like to stay in San Francisco, rather than transfer to Los Angeles."

But I wouldn't say to my good friend " I want to remain in the car when you get to the store."
That would be considered very unnatural for familiar speech.