As someone who is a fan of both, I'm becoming more of a Universal fan when it comes to it's theme parks.
Disney has the nostalgia factor as
@RamblingMad mentioned. Which is one of the reasons why I do cherish Disney as a whole, not just park wise. I've always been obsessed with learning it's history, as well as collect vintage Disney items.
However, from the moment Disney announced New FL and Universal announced WWoHP, within the last decade, or so alone since that moment, Universal has dominated in all the content it has released and revolutionized, as well as builds significantly faster than Disney does.
Sometimes faster doesn't necessarily means better, but in Universal's case, unlike some people, or companies who sometimes create something quicker, but subpar, Universal has reached the mark into doing something unique and great, even though on the occasion the newer content doesn't thrill me (one example: FaF).
I feel like Universal really does try to push to be unique, do better, do it's best for attractions, especially within the last decade, where as Disney I feel nowadays only strives and pushes when under competition, or only for certain projects.