In season 6 of friends TV series, episode 2, in 4:03 Chandler says: "I'm gonna be moving out." My question is, what's the difference between that sentence and the "I'm going to move out." Is there any difference between them?
The use of "gonna" is informal speech meaning "going to", so that makes no difference to the meaning.
"I'm going to move out" is a simple declaration of intention. It could be followed by a time qualifier: "tomorrow".
"I'm going to be moving out." is a little vaguer, and speaks of a process rather than an event, so it might take a little while to do it: "over the next week".
In practice, the meanings may be the same, depending on how they are qualified and the intention of the speaker.