I have both phones in my household and I can honestly say the GS3 is a far better phone from a hardware standpoint. Software though, the Nexus wins because of the pure Google experience and ability to root and customize.
Back to the hardware, the S4 processor is a beast and runs much more efficient than the Nexus, leading to a snappy UI and incredible battery life for a Verizon LTE phone. The screen is improved - slightly larger with more accurate colors than the Nexus. The camera completely blew me away. No comparison here, the Nexus is complete crap compared to the GS3. Best camera I've tested on a phone so far. Last but not least, the radios also seem to be much improved for a Samsung phone. Not a single issue so far - in fact I don't think I've seen it slip out of 4g yet.
I would get this phone in a heartbeat, if not for my need to tinker with software. I don't like hacked work-arounds so the locked bootloader is a deal breaker for me. The only hope for this phone is if something useful can be pulled from the Verizon Developer Edition.