I think the character needs the ability to blend into most scenes as a civilian. If Logan didn't have a mask, then his face would be instantly recognizable to the world. A big part of the character's personality is the fact that he is able to blend into a bar or similar scenarios if needed.
The rest of the X-Men don't really need masks for the most part because they don't interact with society by remaining anonymous.
The old Logan Wolverine didn't even know who he was, so I guess the only reason to wear the mask is when he was with the X-Men.