This is the only "fix" that has worked for me even after the 6.0.1 iOS update on my iPhone 4S...
My iPad 2 running iOS 6 did not experience any problems with the router when using WPA2, updates to apps, weather etc.
My wife has an iPhone 4S, I did not upgrade it to iOS 6 once mine had issues, no issues have been encountered using the iOS 5 on the router. iOS 5 worked perfectly using the WPA2 settings of my routher.
I changed my router back to WEP rather than WPA or WPA2. Blam, I can go into the Weather App and it works, oh and I can tap the App Store... Ding, ding, ding Updates actually tells me that all my apps are up to date rather than spin the loading icon. I did find it odd that updates and weather were the main issues. Funny how featured, top charts, email and more worked fine using WPA and WPA2...
Router: Netgear DGN1000, latest firmware V1.1.00.51_NA.
So I attempted:
1. "Reset Network Settings" on iPhone
2. Down grading the firmware of the router to V1.1.00.40_NA
3. Power cycle iPhone
4. Clear the Safari cache
5. Default Reset on Router
6. Changed HTTP Proxy to "Auto"
7. Airplane mode
8. Forget Network on iPhone
I am sure I attemped more, but I can't remember them all. I attempted every option I could find online.
I decided to change from WPA2 to WPA... Same problems with my iPhone 4S.
Changed from WPA to WEP. Fixed all the issues, weather loads quickly, app store updates screen loads right away.
I updated the router firmware back to .51, kept it on WEP and my iOS 6.0.1 iPhone 4S works perfectly. I would rather use WPA2, but I would also like to have the features of my iPhone work correctly.
This may not work for everyone, but I believe my iOS 6.0.1 iPhone 4S is having issues only with WPA and WPA2. I am going to update my wife's iPhone 4S to see what happens with her phone on WPA and WPA2.