Updates on Android
Google Play
Play updates the app like any other; there is no in-app updater on Android. Settings, This app, shows the version and "Updates arrive through Google Play". Turn on auto-update for the app in Play if you host: a host on a newer version may not open in an older app.
Sideloaded APK
Download the newer APK from the releases page and install it over the old one. Android keeps the app's data, including your world. The APK and the Play build are signed differently, so switching between them means uninstalling first, which deletes the world; export the characters and workshops you want to keep before you do.
What an update does to your world
Each app version carries a specific server version for the device world. Updating the app updates that server; the database, uploads, generated pictures and a downloaded sound library are preserved across the payload upgrade. The first start after an update unpacks the new server and takes a few seconds longer.
Staying in step
The app draws a host's screens itself when the host runs server 0.21.0 or newer, and falls back to the host's own pages for older ones. Keeping the app current keeps every host you visit on the app's own screens. See Version compatibility and What's new.