Here is an issue with the 360 that I think is a major one, a very stupid design flaw that I still don't understand why they haven't updated it. Anything you buy digitally is playable offline as long as you're on the original console you downloaded it to. You can play it on another console if you download it again or physically copy it over as long as you're connected to Xbox Live.
Here's the hitch, DLC for several games require the title update in order for that DLC to be functional, even if playing offline. The Xbox only saves a certain number of updates to your HDD regardless of HDD or update size. I don't know the number offhand, but let's say it's 10. Once you play that 11th game that requires an online update, game update #1 is deleted no matter how big or small it is. To complicate matters worse you can't just download the update then back it up yourself like you could other files, because as far as I can tell (and I might be wrong) all updates aren't necessarily stored in the normal game folder file structure like other DLC is.
Prime example of this in action is with Street Fighter X Tekken. The console I play this on is the same console I purchased the DLC with, so I can use the DLC without having any internet connection. Fired it up one day when my ISP was down and it kept telling me my DLC was corrupt and some other error messages. Come to find out the DLC requires the newest game update which wasn't on my system since I hadn't played the game in several months, and I was unable to use any of the DLC until I was able to get on XBL. This works the same with Mortal Kombat and other games. There's also games like Black Ops that have important offline gameplay functions in these title updates, so one can't say these title updates are just to fix multiplayer or online issues. Having to download the same update files every week or two is extremely annoying and is something that should have been done away with years ago. If MS ever decides to kill update/online support for certain titles (or XBL for the 360 altogether) this type digital content is rendered useless even if you're playing on the original system you bought it on.