I've only done the Universal parks in Florida and the Disney parks in Florida and Paris but I honestly do prefer Universal. Largely because I think it's a lot more laid back (well at least when I went) and more "adult". At Disney you obviously have got a lot more kids and the parks are far bigger than what Universal have, but also have a lot more to do and you feel like you've got to plan it a lot. The lack of single rider on most of Disney's rides as well doesn't help, but Universal have it for nearly every major one. Even on a moderate day, I did IOA in about 6 hours (including breaks) and did pretty much everything I wanted to.
Don't get me wrong, I absolutely LOVE Disney and I loved the parks in Florida but doing it was a lot more hectic and tiring in comparison. The overall area which it occupies is huge and park hopping, waiting for buses etc took a lot of energy out of me in the hot Florida sun. Universal was better at this since their 2 current theme parks are right next door to each other.
As for the parks themselves, although I had a good time at all 4 of them, I do have to say none of them gave me the impression that they were better than Paris' Magic Kingdom. The Magic Kingdom in Florida has some great rides, but some like Pirates, Big Thunder Mountain, Space Mountain were me enjoyable in Paris for me and the Cinderella Castle didn't impress me as much as it impressed me in Paris. Hollywood Studios was fun, but with Galaxy's Edge and Runaway Railway not open yet, I don't feel I "fully" enjoyed it and considering it opened as a counter to Universal, it just feels like an inferior version of it. Animal Kingdom was beautiful but just felt a bit poorly laid out and the only ride I really enjoyed there was Everest (I didn't do Avatar because quite frankly the film is awful and why I queue nearly 5 hours for both attractions). Epcot's World Showcase was great but I treat it more like a place to relax and the Future world didn't do anything for me either.
I am by no means trying to mug off Disney, I really loved 4 parks in different ways but I honestly don't think for the most part it was the best of the parks I did out there. However some of their rides like Seven Dwarfs, Tower of Terror, Frozen, Everest, Splash Mountain and Haunted Mansion are all top tier attractions.
As for Universal, the main Studios was "okay" in terms of rides but the overall atmosphere was fantastic. It's such a pretty park and with no rush I really enjoyed just taking it easy. IOA though is a different beast, it does brilliantly in BOTH. The majority of the rides there are fantastic and the atmosphere and theming is probably the best I've experienced in a park. Again, like I said it's a park where you can take it easy and soak it in a lot better than you can do at any of the 4 Disney Parks down the road. However I did find a lot of the attractions lacklustre, Hollywood Rip Ride Rocket is terrible and Gringotts is disappointing (great dark ride, meh coaster).
So in short, both are great. Disney are better at the rides and Universal are better at creating theming and atmosphere. But largely for the simple fact I enjoyed it more, I'll go with Universal. Will be interesting to see if my opinions change if/when I ever visit the other parks though.