I also would just like to reiterate that the issue being described by myself and others in this thread is a new issue. The 1004 error is to some extent generic (as many errors are), and this issue is in fact not related to the iTunes on a Mac error or the restore error that has been written about extensively over the past few years.
This error (which happens to express itself as 1004) occurs on the iPhone when trying to download or update applications from the iPhone App store. It started happening just prior to the 4.3.3 update (I had the issue appear out of nowhere a few days before that update). Upadting to 4.3.3 did not fix the issue. It is also accompanied by a strange behavior of not being able to View your Apple ID on the iPhone. You can log-in, but when clicking to try to view account details, you get stuck in an endless password loop. Finally, I can confirm that all the above still works perfectly on my iTouch and IPad under the same account. I also have absolutely no issues with iTunes on my Mac (all three devices also are currently syncing to the same iTunes catalog).
This absolutely seems like a server side issue to me. For those experiencing the issue, can we try to confirm here if this is happening only on the iPhone? I not sure, but I can't recall anyone reporting the issue on the iTouch or iPad (and again I am not talking about the generic 1004 error but this specific issue).