This is actually more complex than you present it. Latin took off in barbaric, uncivilized places, while Greek left its impact in places where empires had already shone and cities already existed. Greek did leave an impact in some places, such as Egypt, where Coptic was not only written with mostly Greek letters, but contained many Greek words (but very little, if any at all, Latin). Furthermore, the Arabic conquests of the area wiped out much of the Greek, that even in modern Egyptian no trace of the Greek can be found, while Europe remained Latin speaking peoples.