The widget and the app should sync your tasks each time you make a change either in the app or on the web-app at todoist.com.
To perform a manual sync, please open the Android app and tap on the circular sync icon at the top. Also, please make sure that you've chosen the correct view in the widget. If you tap on the header of the widget you can select a date view, labels, projects or custom filters.
If a manual sync won't help, please go to Android's Settings -> Apps and clear Todoist's cache.