Okay, for this fly777ual you need to make a few assumptions. One is, that both engines types being compared are roughly technologically equal. Okay the reason why a quad jet in theory burns less fuel over long haul flights is simple. On a quad jet, each engine must have 25% excess thrust. And, on a twin jet, each engine must must have 50% excess thrust. This means, that, a quad jet is operating much closer to its ideal efficieny in cruise. But, it doesn't stop there. The wing structure of a quad jet is simplier, and lighter, and doesn't require as much re-enforcing as a twin jet, as the engines weight are distributed more evenly across the wing, and the extra weight further out on the wing from engines 1 and 4 help combate the wing's upward flex, thus, it doesn't have to be as strong, and can be built ligher and simplier. It really only makes a difference over long haul flights. (you have a higher maintaince requirement, and in some cases the 4 engines total weight is greater than the alternative 2 engines)
BryanG - about A340-200 vs 777-200 fuel burn. That must be on a per seat basis. The A340-200 is significantly smaller than a 777-200 - a lot smaller. Naturally, with size, comes efficiency - so we should expect that the 777-200 burns less per seat than any significantly smaller aircraft. It is interesting that the article did not make a comparison with the larger A340-300 - which, although still smaller is closer to the 772's capacity. Regardless, as the boeing charmain regularly points out, fuel consumption is typically about 10% of operating costs, so, you can have a hudge saving, such as 20%, and that only translates into about a 2% overall operating cost saving. This is why airlines aren't getting too worried about it.