Frequently, it is simply a matter of preference. However, the following variations could be interpreted slightly differently.
I don't know.
A simple declaration. Unambiguous.
I don't really know.
This is often used to deflect a negative self-assessment. It's common to not want to admit that we don't know something, so we occasionally de-emphasize the declaration.
I really don't know.
You didn't mention this variation, but this is often used to do the opposite, actually emphasizing the fact that we don't know. This can be used to stress to someone who may not believe you, that you in fact, really don't know something.
These are really just minor nuances, and the phrases are often used interchangeably. If someone wants to express a degree of knowledge, some other phrase is usually used in conjunction with the declaration. For example, "I don't know enough about X to answer that."
The phrase know which way the wind blows is quite old. Even the sense of "knowing the direction of public opinion" dates from at least the early nineteenth century (see The Phrase Finder).
There is a slight association between that phrase and a "weatherman" that dates from at least the early twentieth century. There was a classroom activity called "Weathervane" to teach compass directions in which one person was designated "weatherman" and called out an arbitrary "direction of the wind" (description).
Beyond those facts, there seems to be little evidence that Dylan's lyric existed as a phrase before he set it down in the Sixties. (It's hard to prove a negative, of course.)
Best Answer
When expressing frustration or exasperation, we often begin by saying something like, "If I had only known..."
Yet, in all these instances, life offers no crystal ball; there's no way of knowing what the future holds, or to be cognizant of something we are unaware of. This is the essence of "You don't know what you don't know," only it's being expressed in a comical way, much like Yogi Berra might have said.
(To those unfamiliar with the Yogi Berra reference, he was a professional baseball player who was reknowned for his wry way of saying things, such as, "It ain't over 'til it's over," and, "You can observe a lot by watching." That said, you couldn't know what you didn't know, but now you know.)
Generally speaking, it would not be a good way to express something in formal writing, unless perhaps you were deliberately injecting humor.