Go back far enough and Disney was the one that actually started the screenz craze in the 1980s with the development of the original Star Tours. According to legend, Lucas took Spielberg to see what they were working on and he went to Universal later and asked why they weren't doing anything like that. Several short years later and Back the Future: The Ride was conceived.
Both companies tend to go through phases. I'd say Disney is just now coming out of their projection mapping phase (after adding it in to every ride and show they could think of, culminating with their master opus Runaway Railway,) and then they'll be diving head first back into full-on screenz for a little while (and coasters... and coasters with screenz). Universal probably wants to get back to opening new revolutionary ride systems that blow everyone's mind, cause it felt good when they opened FJ.