I guess there's some standard questions, eh?
Snapshot doesn't add a file to your desktop - so, can you move ANY file (or add one) to your desktop? A good choice might be to try to add a new folder?
Check in Get Info for your Desktop folder. Is your desktop folder locked? I just tried that, and moving an icon shows the 'not allowed' symbol, and I can't move any icons. That's likely it, eh?
If that doesn't help you -
Is your desktop still locked when you log out, then log back in to another account?