Why can't I see the weather information in Notification Center?
I realize that this thread is old, but there are so many issues with iOS7, that there are probably many who still haven't found a solution to this.  What worked for me: Settings > Privacy > Location Services -  Turn on Location Services (if it's not already), and also, scroll down to Weather and make sure that specific app is turned on.  After that, the text weather showed up under the date on Notification Center.  And I have my Weather app in a folder.

iPhone 4S
iOS7