I think Disney does care, in terms of learning from them. For example, if it is true that the DHS redo will result in immersive themed lands, I think it is the continuation a trend of Disney and Uni continually learning from each other in that regard. Disney started it with MK lands that had looser (Frontierland) or stronger (Tomorrowland) themes pulling the architecture and attractions together. Uni takes the next step in Islands of Adventure, with areas like Dr. Suess and Super Hero Island playing the themes even stronger. Disney does it even better at Tokyo DisneySea. Disney takes the next major step in Carsland, making the theming so strong that almost nothing intrudes, and taking theming into all elements of the experiences (food and drink, shopping, meet n greets, attractions, visual elements, lighting, sounds) so that you feel immersed in the fictional story. Uni goes even further in Diagon Alley. Each advance was successful, and so the two companies learned and kept going further. Disney now seems set (if what we hear is correct) to do something similar in terms of Star Wars and Pixar worlds, not to mention what's happening with Pandora at DAK.
So, I don't think Disney feels like they need to upgrade and update so Uni doesn't take business. But I do think that when Disney decides it's time to upgrade and update, they look at what worked for Uni (and others) and take that into account.