It all really boils down to the thought that they are not competing products.
Boeing believe that the industry will evolve toward a more "point to point" model, requiring an aircraft like the 787 to connect more secondary airports. Airbus believe that the future is hubs, and the airlines will need large aircraft (ie, 380) to connect the hubs. The truth is likely to be between the two positions - there will always be a demand for p2p, and there are many city pairs that are crying out for more capacity, but since they have limited slots, they will need larger aircraft.
Who will win? I reckon they both will to be honest. The really interesting fight is not 380 v 787, but 787 v 350, and 747-8 v 380. My guess is that the only loser will be the 747-8; the 787 and 350 can happily co-exist (cf, 320 and 737). The dear old 747 is getting a bit long in the tooth, and Boeing have found it very tough to sell any pax variants since 380 was formally launched.
All great products - but horses for courses and a lot will depend on the overall shape of the industry rather than the merits of any one type.