Which sentence is correct -- "I have to go" or "I want to go" -- and why?

Both sentences are correct. It all depends on context. You should use “I have to go” when you imply necessity. Example: I have to go the bathroom because I need to wash my face. (It implies that you need to wash your face.) You should use “I want to go” if it is simply based on whim or if there’s no necessity in doing so. Example: “I want to go to the bathroom.” (It only implies that you are a human wanting to use your motor skills.)