The app starts back up because you have it set to update at an update interval. It has to run if it is going to update. It would be nice if it would just open up when the time comes to update, but may I ask, how does the app know it is time to update if it isn't running?
Anyway, just running isn't going to severely drain your battery. I have been using weatherbug for the last month and have experienced 2 times where it was the culprit for draining my battery. I checked my partial wake usage (tells you what apps are waking your phone up and how long they keep it awake for) and saw that weatherbug was going nuts and keeping my phone from sleeping. A simple stop using the built-in running services in settings took care of that each time.
As long as an app is not keeping your phone from sleeping, it really isn't going to drain your battery in a noticeable way.
I have my weatherbug now set to update every 6 hours, allow it to use my location, and allow it to use cell towers. I do not allow it to follow me in the background and I do not allow it to use GPS since I don't leave GPS on. My network location is good enough for my weather. I used to have weatherbug update every 30 mins since it was my primary weather app. Now that it is not and I just use it for detailed weather info (i have a different widget that I allow to follow me), I only update every 6 hours.
Do yourself a favor and download spare parts or some other app that links you to your phones battery history and usage and see for yourself that allowing weatherbug to update more or less frequently isn't keeping your phone awake for long enough to severely affect battery life unless you are letting it track you in the background and/or with GPS. Excluding of course when/if it gets in the wacked out state i described.
What you want is to be able to see your partial wake usage, and like I said, Spare Parts will link you to it. The Battery Status battery widget also has a link. Last, BetterCut will provide an activity shortcut to this usage screen I'm talking about.