Is there a difference between the phrases "You don't appear to know [...]" and "You don't appear to be knowing [...]"? Can every occurrence of either be replaced by the other one without changing the meaning of the sentence?
"Know" is a state of information being stored and recallable in your brain. It is present tense, meaning that you can recall and use the information now if you need to. "Appear to know" or "don't appear to know" is something that can be assessed through a person's responses to a question or situation. Their knowledge, or apparent lack of it, can be obvious through their ability to correctly answer questions or perform a task.
"Knowing" is the continuous form of the word, it describes a more active process. The information is actively in your mind, not just accessible. That's the case when you are actually using the information, but someone else would have no way to distinguish whether it is in your conscious mind because you just recalled it, or it is stuck there like a song you can't get out of your head. There is no way for someone else to distinguish what is actively in your mind on a continuing basis.
I suppose in a unique situation like someone complaining that they can't get a thought out of their head, you could respond that they appear to be "knowing" that information, although that wouldn't be idiomatic. If they weren't providing that context, there would be no basis for you to conclude that they "don't appear to be knowing..."