Just spotted a crucial point: you're flying from USA to Greece, therefore probably via London Heathrow if you're on BA?
The BA First experience on that route - it's the 777-200 - is fine.
When you land at Heathrow, however, you'll be travelling Club Europe for the ongoing 4-5hrs to Athens, which is like economy without someone sat next to you. The planes they use are tatty old ones too IIRC (did it last November).
If you travel as a First passenger you'll get the Concorde Room lounge at T5 if there's time between connections. Club World you won't, but the ongoing Club Europe flight class will be identical.
Tax and Avios will be worse than using AA connecting in the States due to the multiple flights through a UK-based connection too, I'm afraid.