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: 
Well, one must remember that Latin, was the primary and central language of the Roman Empire-(though the Greek language was a close runner-up, particularly among the Nobility and Patrician related classes). Latin's linguistic influence was far more widespread in countries to the North and West of Rome, due in great part to the continuity of the Greek language in lands to the East and South-(though to a limited extent) of Rome. 

Although the Romans had conquered the near entirety of the centuries old and diverse Greco-Hellenistic imperial zones, the strength and continuity of the Greek language lived on in the various academic institutions, though had also continued, in many cases, within the larger institutions of everyday society. In other words, the Greek language, even under Roman colonial occupation, still retained its elite and preferential status, continuity, indispensability.

Comparatively speaking, the Romans had greater cultural freedom and autonomy in the more underdeveloped lands to its West and particularly, to its North. Much of the WESTERN Roman empire was a primitive backwater and thus, for the Romans, it not only provided them with the ability to pave better roads and bridges, but also to pave an uniquely autonomous cultural and linguistic legacy and continuity.
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.