Either way, it should be frowned upon as not a nice thing to do, but would it actually be assur?
Yes, downloading "pirated" music is "stealing"*. Next question, please?
If you are asking about specific instances in which someone had permission (explicit or otherwise) to download music from someone who had permission (explicit or otherwise) to upload it, then you're not talking about downloading "pirated" music.
It is a very subtle difference, but it (potentially) makes all the difference in the world.
Ultimately, Dina DeMalchutha Dina. If you have no right in the eyes of the local/state/federal government to download that music because permission needs to be granted to you by someone who has not done so, then it is "stealing"** if you download it. If the person who uploaded it had no right to do so, then it is "pirated". If neither of you had permission to do what you did/are doing then it is "pirated" music and you are "stealing"* it.
*I'm using the word "stealing" here in the way I understand zaq to have used it in the original question - taking something that doesn't belong to you that was provided to the public by someone who didn't have a right to put it there. However, it may not be Halachically stealing; it might be more akin to the secular "receiving stolen property", which might be different. You might have to return it, but you might not face penalties. Or you might NOT have to return it, EVEN THOUGH you might have to pay penalties. That's a separate question. I think the question is about whether or not this is permitted, not about what the technical definition of the activity is.
**I'm using the word "stealing" here in the way I believe is most appropriate. If you are taking something from a domain in which the person who posted it had permission but you do not have permission to download it (eg., a paid, legal subscription service in which you use a hacked membership ID), this would be "stealing" (in some sense of taking something you're not allowed to take) even if it wasn't "pirated".
There are several reasons why copying tapes could be prohibited.
Igros Moshe ruled that it is forbidden as stealing.