literally, "He/She doesn't change/never changes."
literally "He/She hasn't changed."
I'm not sure where "stubbornness" comes into play, since I don't know the origin of the quote. But essentially you are correct, the speaker is using the subtle difference between the two tenses to make a point about the person. I might translate it into something more like:
"I don't know whether to say he hasn't changed, or he can't change!" ... In English, these are different tenses than the Japanese, but the sense of the sentence might be conveyed more faithfully.