What's the difference of divorce final and divorce closed?

I am not certain that I fully understand your question, but I will give it a try. A final divorce (or dissolution) means that the judge assigned to your case has signed, and filed the document that constitutes your decree of dissolution. This may be a separate document submitted to the judge for signature, or it may be a minute entry, that is a notice from the court, that the judge has stated to be the decree, and has signed and filed. A closed divorce could be a divorce case where the time limits imposed by the court under the rules have not been met, and the case is therefore dismissed. Another meaning might be that the final decree has been entered, and the time for an appeal from the decree has run, so therefore the case is closed.

I'm not exactly sure what document(s) you may be looking at where you saw these terms or what you were maybe reading online, but a final divorce decree/judgment is the actual document (court pleading) signed by the judge that grants the actual divorce such that the parties are no longer married. If you are unrepresented, this is typically done through a court hearing that you would need to attend. You of course have to first file a lawsuit for divorce to take certain steps and file certain paperwork to get to that final part. Likely, divorce closed is simply the clerk/court officially closing a divorce file once it is over, or if the file was open for too long and never processed, closed due to inaction. But it may be best to just consult with an attorney in person to review your actual documents or if you have a pending case going on that you are unsure of the status or next stages.

You are asking as question in an Illinois forum. There is no such distinction in Illinois.

Probably that the divorce final is when it was granted, divorce closed is when the court sends the file to storage.