Any widget that is configured to poll at a certain frequency has the potential to drain the battery.
1) How often does it poll for updates (I haven't used the one in a while; I forget what the options are)
2) If you install spare parts, you can check partial wake usage since last unplugged to see if it's holding any wakelocks (it probably does briefly, but you can see if it's longer than normal). To do that, install spare parts, plug the phone in briefly, unplug it, let the phone sit for a few hours (making sure weather updates), then open spare parts, select battery history, and check partial wake usage (top dropdown) since last unplugged (second dropdown). If you see it high up on the list, it could easily be draining the battery a lot. Also, check network usage and see if it's hight on that list as well.