I would choose the S3. It tops the One X in almost every feature, except probably the casing. The casing of the One X definitely looks better, but for me the looks of a phone are not decisive for my choice.

I like the S3 better because it has a bigger battery which obviously lasts a lot longer and is also replaceable. The screen is supposed to be better in the One X, but I doubt that's noticeable.

To be honest: my true choice would have been a Galaxy Nexus. It seems to be the perfect phone to me, unfortunately it's got one major flaw: the volume and quality of the sound. Since this is a PHONE, this is a deal breaker for me. The volume of my current phone both for ringing sounds and normal speakers is too low as well and I resent it for it.
So here's hoping there will be a new Nexus phone soon. I don't really care if it's quad core or not, but I would like to see it use ARM's Cortex A15 instead of Cortex A9.