Upon re-reading many of the well thought out inputs here, I have to agree that comparison between aircraft such as the B777 and B747 is unfair, the B747 does not have the advantage of the much later refined super-critical wing of the B777, and has good, but older technology engines. The only fair comparison would be to build a 2 engined B747 with comparable technology engines, or alternatively to build a 4 engined version of the B777 with comparable technology engines.
Fortunately for this argument, Airbus has done just that for us in building the very similar technology A330 and A340, and CONF iture's post has well illustrated the fuel efficiency differences between the essentially similar 2 engined A330 and 4 engined A340.
It all gets down to excess thrust allowing for much improved climb, and the ability to cruise at higher levels at all phases of flight at similar weights.
Regards,
Old Smokey