Welcome
I'm afraid it's not simple. Some apps store data files on SD (or the internal storage emulating an sd card), but in general an app's internal data is stored in /data/data/<app package name>. You need to be rooted to look at that area, and moving apps to SD does not move that. Since you mention a weather app, it's common for such an app to store graphics it uses for the widget on /sdcard, but the current weather data will be in /data/data. It is however possible that the whole lot is in /data/data.
In fact when you say "I know I can move an app from internal to external" that raises questions, because since 4.0 most devices
don't allow that (it was only standard in 2.2 and 2.3, though some manufacturers added it back for some of their devices, and it only ever moved part of the app anyway). Also moving apps with widgets to SD (i.e. real SD, not the emulated "internal" sd) will break the widgets - they may work for a while, but they'll fail the next time you reboot.
As for the difference between apps and widgets, they are not really different things. A widget is a live or interactive image on the homescreen, showing live data provided by an app (a clock, weather display, calendar, etc) or allowing interaction with an app (e.g. controlling a music player). Some apps do nothing except provide a widget, some do not have widgets at all, and some apps have a "normal" app and a widget (e.g. a calendar app may provide a widget which lets you display your calendar on the homescreen, and if you click on that opens the calendar app, but you can also use the calendar app even if you don't use the widget). So a widget is an interactive or updating tile, as opposed to a static icon, provided by an app.