You obviously have narrow tastes in "a better picture". Good for you.
Narrow tastes? Nah, just a better understanding of what the features do apparently (or actually, what they DON'T do on these camera phones). I have a more realistic outlook on what both phones are capable of.
Bottom line, both of them suck at taking low light pictures...regardless of options or settings...they are also inadequate at taking good action shots because of the slow shutter speed (again, regardless of features...even the so called "Burst" mode on the GSIII which is laughable).
Both phones are more convenient point and shoots...nothing more. The iPhone just takes better pictures than the GSIII, with or without customizable settings...especially in optimal environments where these camera phones really shine, outside of that, they are kind of useless as far as "good" photography.
It's ok though, you have a lower level of knowledge on it, i don't blame you for that.