Widgets draw power based on what they do, just like any other app. If a widget just shows information from the phone, it doesn't draw much power at all. If it checks the Internet continuously or runs some intensive process to generate whatever it is showing, it can potentially draw a lot of power.
Most weather apps/widgets use location services and data during the update process. They don't use much otherwise if they are well-written and use a decent ad service (or no ad service at all, if it's a paid app). Set your weather app to update itself, say, every hour or two and you shouldn't see much difference.