Downsides of distro pacakges:

someone needs to package an application for each distro
applications often need to maintain support for multiple versions of some of their dependencies to be able to continue to work on multiple distros
users of different distros use different versions of the application, creating more support work for upstream
users of some distros can’t use the application at all because there is no package
adding 3rd party package repos is dangerous; every package effectively gets root access, and in many cases every repo has the ability to replace any distro-provided package by including one with a higher version number. 3rd party repos bring the possibility of breaking your system through malice or incompetence.

Downsides of flatpak:

application maintainers are responsible for shipping shipping their dependencies, and may not be as competent at shipping security updates as distro security teams
more disk space is used by applications potentially bringing their own copies of the same dependencies

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