Always with the hard copy. I like my games collection... and it wouldn't really be a collection if it was all downloaded. I do have some digitally-distributed PC games, but only those that are hard or impossible to get a hard copy of (except for Fallout 3, which I won in a competition).
That said: I used to feel exactly the same about CDs - I have over a thousand at home but it became completely impractical to have that much stuff laying around when I went to university. I soon got an iPod to replace my old music player, so started buying from iTunes rather than physical copies. Maybe I'll eventually go the same way with games if there comes a time when there's a well-stocked digital distribution system for Xbox 360 - the current Games on Demand thing is... well, a token effort at best. Of course, price is also a factor.