When you develop and run an application locally, you are actually signing it with a debug key.
When you export, you are signing it with a different, production key.
Google Maps requires your api key to match the key you sign your app with. So you will need 2 different api keys depending on whether your apk was made via debug key (copying it from "bin") or production (exporting).
What's happening to you is when you export your app, the key is not matching what Google Maps API thinks it should be, so it's not working.
Read more about signing apps here.
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