AccuWeather struck a deal with MANY phone manufacturers, so they license the Weather information and basic mainframe for the app, but the overall app design and widget look is, ultimately, in the hands of the manufacturer. So, even though it SEEMS like AccuWeather's widget is in LG's phone or in Samsung's, it's really a skinned version that each manufacturer has come up with (sort of like launchers and Android in general; the basic stuff is there, but every handset maker tweaks it to their liking).
You COULD ask someone to rip the app from LG's phone, upload the APK, and sideload it onto your S4 or try AccuWeather's app from the PlayStore (THAT app is sure to look the same in any similar phone in terms of screen size/resolution).
12-03-2013 10:58 PM