Both phones have good displays. While iPhone display has somewhat better specs in some categories, they are close for the most part. There is only one category where the difference in the specs is huge. It's a contrast ratio. Here SGSIII is still better almost two orders of magnitude (45,000 vs 1,356). Nothing surprising here. While one can argue which colors are more accurate, it's hard to argue that display without black color (iPhone) is really competitive.
Also, some of the claims in DM review are border line ridiculous. For example, they say this about SGSIII "Gamut Too Large 139 percent of Std (sRGB / Rec.709 is the color standard for most content and needed for accurate color reproduction. Note that Too Large a Color Gamut is visually
worse than Too Small.)"
That's just plain incorrect. All people who do photo editing know that wide color gamut monitors are much better (and much more expensive). Wide gamut means that display can reproduce more colors. How to use them is a different story. Some monitors provide special mode (narrowing the gamut) to render sRGB content when used with applications that can't handle color profiles properly.
Besides, the talk about accurate color reproduction on phone displays in general is not than meaningful given the fact that apparently none of the Mobile OSes does color management. Those interested can get more information
here.
