As far as originality goes, I don't think either studio really suffers from a lack of it per se. Pixar's movies often tend to come from pre-existing concepts, and then doing something with the concept. E.g.:
-Toy Story: Toys that are alive, and can't act with humans around (pre-existing) + what happens when a toy doesn't know it's a toy? (Pixar)?
-A Bug's Life: The Ant and the Grasshopper (pre-existing fable) with more complexity (not that it's that complex a story admittedly).
-Monsters Inc.: Monsters live in a children's closet (pre-existing) but are actually scared of children and whatnot (Pixar)
Now the "Pixar" bits aren't actually confined to Pixar, but I feel it's an example of how Pixar generally works. Take a broad theme, then add something specific to it. There are exceptions though (e.g. Up, Brave), and sometimes what's added isn't that deviant (Cars is a story told with...cars, the Incredibles is...a superhero story), but overall, I feel the pattern has generally stuck.
Dreamworks is more willing to do adaptations (e.g. Shrek and HtTYD both came from pre-existing children's books), but that aside, I think Dreamworks's approach is more based on subversion. Shrek is the golden example, to have a setting/story based on fairy tales, but have an ogre be the hero, the princess NOT fall for Prince Charming, sattarize the genre as a whole, etc. Megamind is a story based on role reversal. Over the Hedge (another adaptation, granted), has animals breaking and entering - yeah, the humans cleared out a lot of their forest, but this isn't Bambi). The animals are most certainly not the victims, when a story could have been told where they were quite easily. Even Prince of Egypt is arguably a subversion, in that it takes a pre-existing story (Moses), and focuses mainly on the Moses-Ramses dynamic (which, IMO, is why the film is as good as it is).
Granted, the above is generalizing, and there's exceptions for each studio, but I feel it's a divide that more or less distinguishes them. Pixar will take an idea, and flesh it out. Dreamworks will take an idea, and undercut/subvert/sattarize it. Both can work of course, but the net result is going to be on a "by film" basis.
As for Antz vs. A Bug's Life, looking it up on Wikipedia, there seems to be a lot of contention as to who stole what, and I'm not even sure if it matters. As stated, A Bug's Life is based on a pre-existing fable, and while parallels can be drawn between the two films (both feature a social outcast ant who wants to woo a princess, and must lie to maintain a charade that they've unintentionally walked into), the tone and style is otherwise different. Personally prefer A Bug's Life, but I think Antz is still a decent entry in Dreamworks's lineup.
