It’s very simple. The parts of the former Empire that now speak “Latin” languages—France, Spain, Portugal, Italy, Rumania—were the parts of the Empire in which Latin was the lingua franca. Further east, the lingua franca was Greek, but Latin was still the official language of imperial documents and the courts.
I thought Greek was the language of trade — of business, broadly — all around the Mediterranean. Not sure I ever read why . . .