Heaviest Boeing vs Airbus aircraft: Why large (150 ton) difference in Weights?
Short Answer: 

Difference in corporate mindset between the two companies and incompatible interpretation of market conditions and projections of future market demand. 

Long Answer:

@Anilv answers your question to an extent but you should really understand the total scope of the commercial aviation sector. Airbus is a relatively newer company that has a widely different mission goal than Boeing: Airbus is an exclusively EUROPEAN owned multinational company with a majority share under EADS, the European Aeronautic Defence and Space Company. It was primarily brought together to give Europe and its smaller more specialized companies another chance against massive, government backed corporations from America (the big three: McDonnell Douglas, Lockheed Martin, and our big daddy Boeing). This actually caused quite a bit of a political sh*tstorm, with Boeing complaining that Europe's explicit backing of this new Airbus being far too pointed in corporate tie-in with government, but besides for some concessions made to ease relations with America, Airbus was allowed to continue receiving grants and other national backing from its parent countries. 

Now to explain specifically why Airbus doesn't compete directly ALL the time with Boeing's aircraft is because frankly, leadership of both companies cuts projects all the time in different sectors because of their individual personas and visage for the companies' futures. I'm glad you mentioned the A380 because except for a few other situations continuing the A3XX (in-house title for A380 prototype) was by and large the most controversial internal decision made by any Airbus executive. The situation was this: Both Boeing and Airbus were looking at the VLA (Very Large Aircraft) concept, an idea becoming popular in the 90's due to burgeoning flight routes and demand for efficient carriers of mass amounts of passengers. Also, as the Gulf War ended Airbus had begun to reap profits with their A330 and A340 series aircraft. However, not too long ago Boeing had released its jumbo-jet 747-100 and -200, which were extremely popular and Airbus leadership was getting worried about their potential loss in this possibly goldmine niche. Alan Boyd, at the time president of Airbus North America reported after a joint study with Boeing of the market that Airbus predicted a market for about 500 to 600 airplanes, while Boeing envisioned a market for only 300 to 350. Interestingly both saw a profit margin of about $15 billion. Boeing saw the risk/reward too high, and at the time (I would argue it still is) was very encumbered with bureaucracy and promises to keep to the American government. It was a large project no matter what, and Boeing to an extent was on America's payroll after all. However, although risks were high for Airbus as well, the company exec at the time, the legendary Jean Pierson saw that the risk was worth taking, not only because of projected efficiency and prototype designs, but also because he needed to protect the rest of Airbus' lineup. 

To even comprehend the complexity of the issues facing these men, you MUST understand that decisions to make or scrap a new plane isn't an isolated choice: an airline buying a new plane from one maker is highly unlikely to mix and match to buy a different size plane from another maker just to fit a smaller or larger route i.e. buy the A380 for its long-haul high-demand routes and the 787 for its longer but lower-demand routes. Boeing on the other hand had a relatively fleshed-out spread of planes and due to a HIGHLY incorrect mindset of complacency and EVOlution rather than REVOlution, they did not seize this oppurtunity to expand and improve. This was common knowledge to all business-minded men in the sector who were worth their jumping beans (sadly few at Boeing after , so the most prominent factor in Pierson's mind was probably his defense of the also-new A340 series, a small long-haul aircraft that fills a similar role to the new 787. This, as the chart you provided shows, filled a spot that was projected to be VERY profitable (as Boeing found out too late and rushed in the 787 to fill) but at immediate time was a little too hard to swallow for most medium sized airlines. Keep in mind that Airbus was still trying to prove itself to the world. 

So, by Pierson's logic introducing a superior 747 in the form of the A380 would not only draw attention to the plane itself, but also help establish the A340 and start the ball rolling for future development of the wide-body long range medium size airliner. So, that's pretty much why the A380 was a (mostly) no-brainer for the Airbus group, and from the late 80's on, why Boeing and Airbus' line of airliners has considerably diverged. Hope this answers your question, and if you'd like more information or just clarification feel free to comment. 

[Edit]
The book Boeing Versus Airbus: The Inside Story of the Greatest International Competition In Business gives a wonderful account of everything I just detailed and more. I've read it multiple times and I never cease to learn something new from something under 300 words. Give it a try if you have time!