most of the weather apps starts a service in the background to download data from their servers. Some of them download (Videos, news, ads and weather data) regardless if you are placing a widget on your home screen or not. Widgets are only using the data that is downloaded by other services. Ads from the other side are only downloaded when you open a regular application unless the developer of the app download one or more ads to display them in various locations when downloading weather data. The size of ads is usually 100kb or less which means you will download 5 mb a day if ads are downloaded every half an hour.
If you want to check what data has been downloaded in your device, turn of the internet and use your app, if the app is still functioning and you can view weather data, ads, videos, weather perception map,,,etc, then everything is downloaded in your device every half an hour or whatever you configure the application with.
It is more profitable to download and display ads when a user use the app.