For the Play store, the developer provides a compiled version of the program.

For F-droid, the source code must be available online, and F-droid compiles the program, not the developer. This way you can be sure the source code is the actual code used.

Usually the two versions are the same, but sometimes the f-droid version is the play store paid version, but for free.

Nothing they’re are just too different app stores. F-Droid has more of a focus on FOSS apps and the play store has more of a focus on making money

The fdroid version of Firefox, fennec, has some of the beta features enabled, such as full add-ons support and a few others I can’t recall off hand. I think it removes some telemetry as well.