It's true that Greek was spoken in a large area at some time in human history and indeed it's now spoken mainly in Greece in a mostly uniform way. The main reason that there are not many Greek-deriving languages is political.
Greek was spoken mostly in the eastern part of Mediterranean sea, Balkans, Anatolia (now Turkey), now Syria, Jordan, Israel, Egypt etc even in some parts of Italy where there is still a small bilingual community in south Italy. The speakers of these areas often understood, spoke or even (in a less degree of course) wrote in Greek but the degree Greek was their mother tongue is debatable. Greek was undoubtedly the "lingua franca" of the entire region and this gave it some extra bonus for people to acquire it and use it. But on the other half this also meant that since it was only lingua franca for a large proportion of the population, when the condition changed then other lingua francas took its place. This means that the transaction from lingua franca to mother tongue (even for bilinguals) is a process that apparently did not occur soon enough. The reason for the latter are political.
Let's check the regions where Greek speaker existed at some time:
- The main political power of most of these areas before the Arabian expansions was Byzantium that was mainly Greek-speaking in a large degree. Of course not all provinces shared the same degree of Greek native speakers.
- South Balkans and Anatolia were probably at a larger degree native speakers than Egypt for example. This can be explained partially also by the fact that they remains under the political power of a Greek-speaking state (Byzantium).
- The Anatolia region were not conquered by Arabs (although at some part it did) but by Turkish-speaking people in a process long enough that ended in the 15th century. This can be seen today that the Turkish-Arabic language limit is the Anatolia border. During this era a gradual Greek abandoning began that ended in the 20th century forcibly with the departure of the last remnants of native Greek speakers from Anatolia.
- Many areas that are now Arabic speaking regions like Syria, Jordan, Egypt etc had a quite large spread of a religion diversion from the official Byzantine faith (they are called heresies by the official Orthodox state of Constantinople -now Instanbul-) like Arianism. From this variation came the Coptic Christian of Egypt. The main issue here is that people became estranged by the official authorities of Byzantium and so when the Arab conquered the area they more easily abandoned the old political power (Byzantium) and the language it used (Greek).
- The Balkans have a lot of invasions of Slavic people which where not Greek-speaking. These make a difference since they permanently settled in the (often deserted by wars etc) areas and assimilate with the locals. So the people used a mixture of languages with some being mother tongues and other just lingua francas.
- Regions besides the (unstable) borders of the Byzantium, present day Iraq, partially Iran, Pakistan and a small part of India, can be considered cases where the native Greek speakers were not the majority at any time and so with the political loss of power they eventually assimilated with other speakers. For example the Indo-Greek Kingdom which was a Hellenistic kingdom felt even at its political peak the need to make its coins bilingual. That signified the idea that people in charge felt the Greek language is not illegible by a large proportion of their citizens.
This explains the spread of Greek language in the area where at some times where spoken.
As for the part about the variation, I guess this also has to do with political issues: consider the language the Italian speak now and the language they spoke 1 1/2 century ago: there were various and more widespread dialects while now many exist but fewer in number and spread.