Has Griffith always been refered to as Falcon? I thought it was Hawk. And when did Band of the Hawk become Band of the Falcon? Is just a translation thing or did Miura change it at some point?
After discussing it a bit and Googling and AskJeeving like any GOOD internet citizen would do, I discovered a number of enlightening things.
What's the difference between a Hawk and Falcon ?This website answers this question. The differences are subtle, but physiologically substantial. In my own words, a Falcon is more aesthetically beautiful than a Hawk. Each are Birds of Prey, however the Falcon is more curvacious.
The change in terminology (as evidenced by the kanji TAKA, previously translated as Hawk, but with the added furigana, now translates to Falcon) could represent the metamorphosis of Griffith from human to "god". His appearance, like his bird associate, recieved an aesthetic makeover.
But really... I think we need someone who has both translation skills and knowledge of Birds to give us a definative answer.