Pros and cons of each?
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Google play has auto update, whereas f droid has semi automatic updates where it takes more effort.
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The Google play flavor comes with support for FCM (i.e. push notifications via Google infrastructure). The f-droid flavor relies on regular polling for events (so that messages might come in a little bit later, though I am not sure whether it can keep a connection open for real time push notifications).
Obviously the gplay version is more convenient and also updated more quickly, however the F-Droid version is compiled by a trustworthy 3rd party, so you can be pretty sure that what you get is really based on the officially available source code and does not include any hidden modifications. Depending on you “paranoia” level that might be relevant to you or not 😅