It's not equivalent. In your analogy, the cops have said, if effect, that even though you're breaking the law, it's a nonserious breach and they're willing to look the other way.
In the case of file downloading, the cops, namely the record company copyright holders, are saying it's a serious violation and that they're coming after you. You have the comfort of knowing, however, that the chances of being caught are very low as a practical matter.