the difference between

When you say "I'm living in Paris" it implies a connection with "now". For example, it could mean that you are temporarily living in Paris, or that you recently began to live there.

Examples:

Tom was born in Paris and still lives there. He would say "I live in Paris."

Fred was a friend of yours who lived in London, you ask him how it is in London and he might answer: "I am living in Paris now."

Sam moved to Paris but will return to London in a few months, he might say: "I'm living in Paris but will return to London in a few months."