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‘Forever’ is much more dramatic - it means eternally, till the end of all time.
‘For good’ is usually used in the sense of everyday happenings in life. It would mean permanently, for the rest of my life.
“We can’t stay here forever!” - we have been waiting much too long.
“I quit smoking for good two years ago.” - I haven’t smoked for two years.
“The store closed for good last week.” = The store is closed permanently and will not reopen.
“I will love you forever.” - no guy in his right mind would just say “I will love you for good.”
It is a question of the emotional weight you want to put on something. Many times, the two may be interchangeable, though.