'Tell' includes the idea of giving information. The person you tell didn't know it before. So it's not appropriate for greetings and the like:
She said to him, 'It's a nice day today.' [she was just making friendly conversation]
She told him, 'It's a nice day today.' [perhaps was he stuck indoors and couldn't see the sky]
As far as I understood from the link above:
"say goodbye (to somebody)" is correct,
"tell somebody goodbye" is incorrect.
But here is a song:
Listen to Your Heart (a song by Roxette):
Listen to your heart when he's calling for you
Listen to your heart, there's nothing else you can do
I don't know where you're going and I don't know why
But listen to your heart before you tell him goodbye
How to explain here the presence of the incorrect phrase "tell him goodbye"?