A few days ago I noticed that the current temperature in the Today widget in Notification Center was wrong. The Weather widget said it was in the 70's, but the Today widget said it was currently in the 30's.
I did some googling for solutions, and while I didn't find that specific problem, I found various suggestions to remove the widget and add it back. I did that, and now the Today widget simply doesn't show any temperature at all.
I also tried the suggestion of unchecking and checking "Weather" in System Preferences -> Security & Privacy -> Privacy -> Location Services.
The Today widget is in the Notification Center, and it has other information like upcoming calendar and reminder events, just not today's weather.
UPDATE:
The weather information has returned to the Today widget. I rebooted a couple of times over the weekend — once because my battery ran out, and later because I was installing Security Update 2007-01; maybe that triggered it. However, I don't think the current weather is correct. It says it's mostly cloudy, but it's at least 90% clear, and the weather widget confirms that it's sunny. weather.com says Sunny, winds WNW 10-15 MPH, not NW 17 MPH. And it doesn't include the temperature at all in the Today widget, like it did before the original problem.
Best Answer
I think I've found the problem. The weather in the Today widget is only correct if Location Services is able to determine the location, which only happens if I enable WiFi. I normally have WiFi disabled because I use an Ethernet connection to my router.
Temporarily turning WiFi on fixes the weather in the Today widget. The Weather widget doesn't require this because you configure the city explicitly.