In a wider context, there are advantages of both having your passport on you at all times and having only a photocopy. The latter is obvious - you are more prone to lose it than in a hotel safe. For the former, there are situations, where you just can't get back to the hotel to fetch it and you need to leave NOW. I had a medevac in Moscow where I learned this. Even though these are rare, they still happen.

Even when not carrying the passport on me, I like to have some ID - I really don't fancy being a John Doe for days on end. The level of ID I'm ready to show/part with depends on who asks. At an office reception, a DL is more than ample to show/exchange for a visitor's pass (a library card would do, too if only they had pics) If someone in authority needs to establish my identity, the DL is enough. Now if they need to establish my nationality, in most case it isn't, an ID usually is. And when immigration comes into the picture, passport is probably the only accepted one, with the proviso that many EU ID cards are valid for travel in quite a few non-EU countries, too.