I think Pixar is the clear choice.
On the Disney side:
Good films
We all grew up with Disney
Created the genre (yes, there are other animated films, but Disney created the mold everyone uses)
Visually stunning
On the Pixar side:
Can't-leave-your-seat stories (At least most)
Perfected and continues to push the envelope for CGI
Created characters that we care about
Is equally entertaining for both adults and children
Able to be watched 20 times in a short period of time (which is important if you have small children)
At times, side-splittingly funny
While I must give a nod to Disney's ground-breaking, and I will say that in a hundred years some of Disney's movies (like Fantasia and Lion King) will still be loved, I think Pixar is the better overall choice. I wonder if for some, choosing Disney is simply nostalgia love rather than a true comparison of movies. Not for all, but some.
My list:
Wall-E
Finding Nemo
Toy Story
Fantasia
Toy Story 2
Monsters Inc.
Alice in Wonderland
Ratatouille
A Bug's Life
Pinocchio
Dumbo
Snow White and the Seven Dwarfs
Peter Pan
The Incredibles
Cars
Bambi
Lady and The Tramp
Cinderella