Sometimes the background tasks reach their limit and they just stop updating allmodcons.
Pop into settings / applications / background tasks and see if they are set to on or off.
Try turning off any apps you have listed that you maybe don't need and try re enabling your weather apps if they are set to off.
Also just try turning off then back on again any live tile on / off settings in the apps.
Sometimes uninstalling and reinstalling cures this problem too.
Oh and welcome along to WPCentral allmodcons. :)