Incidentally the binary release works much better with the itch app on Linux.
If you want to, you can transfer your saves by copying the save directory from one game directory to the other. By default, the flatpak saves are at:
~/.var/app/com.oddwarg.SulphurNimbus/sulphurnimbus/save
and by default the itch app saves are at:
~/.config/itch/apps/sulphur-nimbus-hels-elixir/Sulphur Nimbus/save
where ~ is your home directory, as is normal. The directories starting with a period (.) are hidden in most file managers. You can likely toggle their visibility with the hotkey Ctrl-H.
If you downladed it without the itch app, then the save directory goes inside the game directory wherever you extracted the archive.