Keeping that in mind, these are two drastically different devices for different reasons. Are you planning on keeping it stock? Are you planning on rooting it?
Without a doubt, if you plan to root or do anything related to that, the Nexus is the only choice here. I cannot and will not ever recommend the Verizon GS3 because the future of it is so uncertain with the locked bootloader.
If you're going to keep it stock and just want the best all around device, then the GS3 is the best choice. Hands down, it has the best hardware on Verizon. Best processor, best camera, best screen and best battery life (besides the Maxx, but that doesn't count). I can speak from first hand experience - I've played with my wife's phone quite a bit and the entire package is very impressive. Battery life is phenomenal, performance is incredibly smooth and the Touchwiz interface is a decent representation of stock ICS. The camera flat-out embarrasses my Nexus
Keep in mind I'm a huge fan of the Nexus - the GS3 is the only device I would recommend over the Nexus...at this point.