It seems to me whils not waiting in line at the supermarket that Disney and PXAR are like a see saw, when one is up, the other is down. This of course is not a hard and fast rule, and there are exceptions, for instance PIXAR released their first film in 1995, when Disney was coasting through the ninties, high on the Disney Renaissance, nonetheless the film was critically acclaimed. PIXAR animators recently spoke about the alleged rivalry between the two studios, passing it off as being healthy and necessary.
The ninties represented the height of disney animation, during the early ninties disney paid PIXAR, then a struggling graphics studio 10 million dollars to co-produce a piece of software with them that would allow them to paint the films using a computer, and bypass the ink and paint stage, when a deal could not subsequently be reached between the two companies, Disney gave PIXAR millions of dollars to produce their first animated feature film, it was called Toy Story. The relationship between PIXAR and Disney subsequently evolved with Disney buying the company in 2006 for 7.4 billion dollars, even more than they paid for LucasFilm. Disney's string of commercial and critical successes ended in 2000 with the release of Fantasia 2000 and Dinosaur, the first films in a long slump for the disney studios commercial and creative output, culminating with disney shutting down it's hand drawn animation department.
For contrast the 2000's represented the roaring twenties for PIXAR, producing a long string of critical and commercial successes. It did seem for a while like PIXAR was unstoppable, after film after film did well at the box office, and with the critics. WALL E, Ratatouille and UP were a string of particularly acclaimed film, coming one after the other. During this period Disney animation was in a slump, the company had failed to make the jump between hand drawn and computer animation, and their films were lacking in the visual and story departments, the box office returns weren't great either.Disney struggled to make the gap between the two mediums, and these films were always lacking visually, the studio underwent some reshuffling after the release of The Lion King, with Jeffrey Katzenberg relocating to Dreamworks for co-found Dreamworks Animation, thus the films that were released during the early 2000's were really the first to be developed after the departure of Katzenberg, a similar effect can be seen now with PIXAR, as the PIXAR films being released now are the first developed since the purchase of PIXAR by Disney, which perhaps explains their problems.
The period in which we are now living is known as the Disney Revival, a series of films higher in quality than those released in the early 2000's. In contrast, PIXAR is suffering what can only be described as a slump, since Cars 2 none of their films have reached the heights of PIXAR's golden age, this can be explained by some reshuffling having taken place since the companys acquisition by Disney.
What is interesting about the alleged rivalry is that each division tries to outdo the other on their own soil, and yet PIXAR rarely comes out on top, in 2007 Disney released Meet The Robinsons, about a boy inventor who travels forward in time, despite not being a huge critical or commercial success, the film is one of my favorites. The movie does tend to encroach on some PIXAR territory, but they do really well with it, and the film is a fun and touching father-son story, in addition to being a sci-fi movie and a buddy comedy. In 2012 PIXAR released Brave, an original fairy tale about a mother daughter relationship that goes awry. Despite not being a Cars 2 style faliure, the film just doesn't match up to Disney fairy tale adaptations like Beauty and the Beast, The Little Mermaid or Tangled, it pales in comparison, and although i have seen it one, i have absolutely no desire to watch it again, whereas i have seen tangled dozens of times, the PIXAR style of animation also doesn't translate well to fantasy stories, and the film doesn't have the painterly style of Tangled of the upcoming Frozen.
This is a tough post to write, as i truly do love PIXAR films, and i am hugely excited about their upcoming attractions, however the Disney Animation Studios is simply more versatile, and has a longer history, and they too have had their slumps, in fact from Sleeping Beauty to The Little Mermaid was one long slump. Overall these slumps are part of any studio, and this doesn't mean that PIXAR had 'had it' or that they're 'over', it simply means that their evolving into a full blown studio, and that this is a hugely exciting time to be a Disney animation fan, because when Disney nails it, they really nail it!