Haven't toyed with the newer 11s Shimanos But had extensive use of the older 7800 and 6600 series.
Used the 7800 from end 2003 till 2009. Used the 6600 for 2 whole years in between for a second bike.
As 'NEW' what's been said above is more or less in the ball park. The differences manifest after about 1 year or maybe 5~10,000km of usage, depending on how hard you ride or how well you maintain the system.
i) The shifting quality of the STI levers is quite evident. Give me a 10 set blind test and I can assure you I can discern the differences with 100% accuracy. The crispness and 'feel' of the DA is obviously on a higher level. But that being said because I had used the 7700 extensively prior to that. My friends who had been exclusively using either 105 or Ultegra previously also could tell the difference but they classed the differences as 'subtle'.
ii)The finishing on the Ultegra derailleurs, cranksets, brakesets and and shifters will start to tarnish after a year or so, even with regular cleaning. The DA ones with normal cleaning and wiping remained pristine.
iii) The shift quality in the Ultegra shifters started to deteriorate once over the 5~10K km mark. The DA one remained perfect for the time of the system I had on my bike, which was almost 6 years. Close to 40,000km.
iv) The Ultegra cassettes were however cheaper and lasted longer. Shifting 'crispness' was marginally different at best but its there. But really very subtle. When you're riding hard and your heart is pounding like mad .... you won't notice it.
So there are differences if you keep the system for longer .. out of the box, not so obvious.
Whether its worth the difference ? Really depends on whether paying that extra is something that doesn't break the bank.