AFAIK, you have to let it download at least the Essential Content, but if you at some point told it to download the full library then your best bet is to just let it finish, as the O.P. didâ€”if necessary via a faster (preferably wired) connection, if possible. Does GB give you a message about missing content? Exactly what do you mean by "a different status"? It's hard to troubleshoot without a specific description of what's happeningâ€”what messages (if any) are you seeing, and if you're not getting a warning or dialog box, how do you know GB is still downloading content? The devil is in the details..
On the other hand, if GB told you the download was complete, but it still says it wants to re-download already-downloaded content, then you could be running into the common issue mentioned in the previous posts. There's apparently still no definitive solution for that, though one person claims to have traced the problem to a particular Project with a missing Loop, so it might be worth it to open GarageBand directly into a new Projectâ€”one that doesn't yet have any Loops or Instruments instantiatedâ€”and see if it still wants to download content. Another possibility might be if your drive is too full, and the download is failing, so it keeps trying (unsuccessfully)â€”again, what messages has GB put up regarding this?
I guess you could always try the solution suggested above (boot up in Safe Mode, shut down, and then boot normally). If GB still insists on trying to download content that you're sure has already been (successfully) downloaded, my next step would be to go to the source (Apple tech support)â€”they should have some experience responding to this issue (since it's so common), and hopefully if you provide enough detail they may have more specific suggestions for possible solutions.