+2
For me (better film bolded)...
1995 Braveheart vs Toy Story- I like them both. Toy Story introduced us to one of the finest ensembles of characters, and to a great new company. Braveheart introduced us to hints of Mel Gibson's insanity. I know which I'm happier to know about. Toy Story is better storytelling.
1998 Shakespeare in Love vs A Bug's Life- A Bug's Life rates at the bottom of my list of Pixar films. It's a lot less adventurous than their best. Shakespere in Love is pretty good too... can't pick.
1999 American Beauty vs Toy Story 2 - Toy Story 2 is more insightful, deeper, more moving, and much funnier and funner.
2001 A Beautiful Mind vs Monsters, Inc. - There are moments of imagination in A Beautiful Mind, trapped within biopic confines. Monsters, Inc. is pure imagination, and heart, and laughter.
2003 The Lord of the Rings: Return of the King vs Finding Nemo- You think you can win this, Lord of the Rings, but you just can't! I just can't put Nemo, Marlin, Dory and co. behind a bunch of people I couldn't care less about.
2004 Million Dollar Baby vs The Incredibles- The easiest choice. I rate the unfeasibly thrilling The Incredibles as Pixar's best film, and the laughable Million Dollar Baby as one of, if not the, worst films ever to win Best Picture.
2006 The Departed vs Cars- Yeah, number 7 on the MoFo top 100 was mad overkill, but it was a return to good if not great form by Scorsese, and though I like Cars, I can't pretend it wasn't minor Pixar.
2007 No Country For Old Men vs Ratatouille- Two great films. Ratatouille is just so damn smart though, saying so much about so many subjects so artfully (it changed how I shop for bread) and Ego is an even better Anton than Chigurh.
2008 Slumdog Millionaire vs WALL-E - Slumdog Millionaire was a blast, Wall-E is half a silent film that recalls the best of that era, half the most fun space film since Star Wars, and all a great love story.
2009 The Hurt Locker vs Up- Up is in the upper tier of Pixar films, and it will always move me, but I rate Hurt Locker as the finest Best Picture winner since Amadeus, so it takes this heavyweight tussle for me.
7-2 for Pixar, 1 even.
So far, I put Toy Story 3 ahead of all the other moviecitynews 'contenders' for Best Picture that I have seen (Social Network, The Kids Are Alright, Winter's Bone, Inception), but obviously a ways to go yet.
Pixar consistently produce better (deeper, surer, truer, funnier, more imaginative) storytelling than the rest of Hollywood. Their works are another level of craft and soul. I wonder if they'll ever be rewarded with a Best Picture win?