Realistically, there are a lot more differences than just the camera. It's not just smaller per se, it's lower spec'd across the board.
For even more information, here is the Axiom compared to the actual
The Axiom and the S3 Mini are very similar. Personally, I won't ever buy anything that isn't either the flagship device, or a variation of the flagship device (Such as the S3 Mini in this case) as I've always been extremely frustrated with lack of support in the development communities as well as from the manufacturers. For those reason, I stay away from all of the other devices. Sadly, the Axiom falls under that category for me personally.
A lot of people (not saying you specifically), buy the cheaper devices thinking they're going to get full experiences and be kept up to date with the latest and greatest and that's just not true. In the end, they look at either the manufacturer or the operating system in general in a negative light, when in reality it was the low price tag. It's akin to buying a Honda Civic because the body might look a little bit like the BMW you truly want. It's just not the same thing.
Anyway, I'm ranting now. For me personally, it's a no-brainer. I'd aim for the S3 without a doubt. That being said, the Axiom is close to the S3 Mini (the true small version of the flagship devices) but there are some distinctions that should be noted.
- Chipset, CPU and GPU are different. This could be due to any number of things (price, bands, availability of hardware).
- The Axiom is on Android 4.0.4. Some might argue how important this might be however, there is a significant boost in performance/smoothness in 4.1 over 4.0+.
- Although neither indicate whether they have Gorilla Glass, it's possible either could. It wouldn't be the first time it wasn't specifically named in the specifications, but was actually used. The chances of the S3 Mini having Gorilla Glass is higher than the Axiom though.
All of the above may not even matter to you, but I have a tendency to dig quite a bit when it comes to my phone and other electronics. I hope this helps in some way!