I won't say goodbye anymore. El uso de la palabra "anymore" indica que anteriormente el orador sí se despedía o decía "adiós." - I won't ever say goodbye. Está claro que el orador no va a decir "adiós" en el futuro, mas no nos dice nada de lo que hacía anteriormente. Además, la palabra "ever" le da cierta intensidad a la aseveración.

Different meanings. I think second sentence's meaning is "I don't say goodbye (never)" First sentence is phrase from Maroon 5 song :)

1st is that i will not say bye to you again .2nd is that i never say bye to y

I think 1st sentence is nonsense, but the 2nd might be said