I'd be willing to bet that Google doesn't have good enough position info from both of you to give you local weather. When you're connected via wifi, Google's location services use your wifi's location as a signal to help pinpoint you.
One way to test that would be to turn off 3G (on izzi's Galaxy Nexus) and the GPS, then connect to the wifi network and launch the Google Maps app. If your location isn't at all accurate on the map, then Google doesn't have a good bead on your wifi's locale. You can then report the problem by following these steps.