The one x is a great handset, the GS 3 is just considered as a more complete package. Both phones are good, they have their strengths and weaknesses.
The advantages of the GS 3 is:
- 1080P recording, better light metering and smoother (but the stereo audio recording sucks, unless it has been fixed in a recent update?)
- removable battery
- SD slot
- better developing community
- audio
- better hardware
- battery life
- better NLED
Advantages of the X:
- design, perceived and build quality. I would say that the X is more durable than the GS 3, as the GS 3 screen seems to crack/shatter very easily if dropped (without a case) due to the screen glass going right to the edge and also the screen seems to scratch very easily, however, as has been said before, it comes down to luck at the end of the day, you put a decent case on either of them, the chances of something like that happening are reduced and a screen protector will protect the screen from scratches without a doubt
- sense (IMO)
Screens are both excellent, comes down to personal preference, many prefer the IPS screen and many prefer the SAMOLED
Software is personal preference as well, only advantage of touchwiz IMO are the tweaks like smart stay, alert etc. (but the majority of these have been ported over as apps etc.)
Most people here aren't bothered about the out of the box software though as they just root and flash a custom ROM like CM or something else based on stock android.
Camera wise, they have their strengths and weaknesses, some shots look better on the GS 3 and some shots look better on the X/S, the camera UI is better on sense IMO and there are far more features as well.
The X did have quite a few issues at launch (but most of them have been fixed with updates and new batches now).
Although you don't seem to be bothered too much about the issues and what most people consider to be cons so just go for what you like the look of the most
Although since size seems to be an issue for you, what about the One S
Better CPU than the X and better than the GS 3 CPU for performance per core, heat and power efficiency as well (would need to make sure you get the S4 snapdragon version though and not the inferior S3 chip), the GPU is considered better than both as well by the hardware expert on here (sarge

), audio is more or less the same as the GS 3, battery life is more or less the same, has a lovely ceramic back (aluminium uni-body micro arc oxidised [for the black model]), lack of NFC (although it may actually have this, however, just not enabled out of the box, not 100% confirmed this yet).
In real world usage, I would say that the S felt more snappier/faster in tasks and the GS 3 feels more smooth overall. The X falls behind both, hopefully with the JB update they will be all more or less the same though, especially the smoothness.
Only disadvantages of the S, is the pentile screen (IMO not really an issue though, the screen is great, only disadvantage IMO is that it isn't as sharp and big as the GS 3 and X screens) and the low internal storage (10GB for photos etc. and 2.21GB for apps)