Some people have asked how an Xbox Live Arcade game decides to play in full mode or in demo mode. The info is actually in the Marketplace FAQ (under Premium Content), but to paraphrase:

To play the full version of any Xbox Live Arcade game that you've downloaded, you must:

• Play it on the console to which it was originally downloaded, or
• If on another console, be signed in to Xbox Live with the profile that downloaded the full game.

If you don't meet either of these criteria, the game will slip into demo mode.

We do this to make sure the game actually belongs to you. You can play it on your friend's console as long as you're there, but when you go home, your friend can only play the demo until they get their own copy from Xbox Live Marketplace.

If you remember, the rules on the original Xbox were even more restrictive; you could only access content on the console it was originally downloaded to. The advent of profiles with Xbox 360 allows for some roaming, while still protecting the content.

I know some folks have run into this after getting a replacement console back from Xbox support. If this happens to you, you can just redownload the game. If you usually sign in to Xbox Live and play, you'll have full access to the game and notice no difference at all. But, if you try to play offline, the restriction will kick in. If you really need to play offline on a replacement console, give support a call and see what they can do for you.