Question: I was just reflecting on the fact that Spanish, Portuguese, Italian, and French all evolved from Latin, the language spread across that area by the Roman Empire 2000 years ago. (Romanian too, I suppose.)

Yet although Greece was firmly a part of the Roman Empire, Latin never took root there, it seems, as the Greek language is quite distinct from the Romance languages. The same is true for the territories above Greece-- that whole region east of the Adriatic aside from Romania.

Why is this so?

Answer: 
Greek states were very much culturally superior to Rome at the time of conquest.
This was recognized even by the Romans themselves. A well-educated Roman had to read and speak Greek. There was absolutely no reasons to introduce Latin in the Greek states. All science, philosophy and much of the literature in the Roman empire was written in Greek.
The old Greek empire, meaning the empire of the Macedonians, the empire created by Alexander spoke Greek widely and had many Greek colonies. The entire city of Alexandria, which was founded by Alexander and became the center of learning in the Western world for 500 years was entirely composed of Greeks and Greek-speaking people. The political control of Rome did nothing to change this.