Considering the importance of Greek in antiquity as the language of both the Alexandrian and the eastern Roman empires, it's odd to me that the modern descendant of ancient Greek is only spoken in such a small area relative to the romance languages. Was Greek never adopted as a mother tongue by conquered people the way Latin was, and if so why? Or was Greek once more widely spoken in the Eastern Mediterranean before being supplanted by other languages?
It has to do with the fundamental question: when X invades Y, which language dominates?
In the case of France, Italy, Spain, in Late Antiquity you have those waves of barbarian migrations, and those cultural groups tended to (a) form societal elites, (b) when settling down they patterned themselves on Roman cultural traditions. They Romanised. This may have been helped by the disintegration of the WRE. It's much easier to be 'Roman' when there isn't an actual Roman Empire around to remind you how un-Roman you are.
Generalising grossly, a conquering elite will eventually adopt the conquered's language if they are (a) a considerable minority, (b) don't have exceptional self-aware identity to maintain their minority language (c) are actually imitating non-native culture and patterns.
What happened to Greek? Firstly, large swaths of what Alexander had conquered in the Middle East only ever had Greek elites. For example, Mesopotamia was largely not speaking Greek at all.
Secondly, the loss of ERE territories occurred differently. Most of it was definitively lost by the rise of Islam and the Arab invasions. Emergent Islam was much less likely to imitate ERE patterns, had a very strong motivator for the spread of Arabic, and was generally pretty successful in cultural transformation. You then of course have the Turkic groups coming in, with some similar dynamics. Basically you have conquerors who extended their language rather than indigenising.
That is the short answer.
Greek was the languege of trade, for a time. Yet we would just found colonies, and not conquer large areas of land, the way Rome did. Because the majority of the "barbarians" in Iberia and Gaul, and the other "barbarians" that settled after the fall of the empire (germanic peoples etc) who became the ruling clashes, adopted the roman languege and customs Latin became really popular. Then you got the catholic Church, who had its writings in Latin. Latin became the common tongue in the western kingdoms of the Middle Ages. Then you fast forward to the colonial era and Spain's dominion over South America. It was only logical that since Spain was such a great power, spanish became a much spoken languege.
Sorry for my sort answer, I would like to write more but it's 1 a.m. here and I'm sleepy.
A followup question: The Byzantines ruled over Greece, Turkey, and much of the Holy Land. If I recall correctly, they spoke primarily Greek. Why didn't those territories speak derivatives of Greek in the way modern Romance languages are spoken?
Eastern Roman/Greek cultural influence isn't so much in the spread of the language, but rather the spread of its orthography in the development of the Cyrillic alphabet, and its propagation in eastern europe, with the Cyrillic alphabet being heavily based on the Greek alphabet.
You have to keep in mind too, one of the key differences between the western church and the eastern church, is that the latin church required their liturgies to be in latin. The eastern church did not make the same requirement. In fact it was part of their early missionary projects to allow their liturgy to be read in the native languages of the territories their missionaries were sent to, which was the driver behind the development of the Cyrillic alphabet, and what eventually lead to eastern bibles being translated into slavic.
And yes Greek was more widely spoken in the eastern mediterranean, but in the middle level of usage by the Roman period. For example, in Roman Egypt, the pattern was imperial and military administration would use latin, provincial administration would use greek, but local administration would use coptic. But by the 9th century, islamicization and arabization were in full swing, and arabic became the preferred administrative language.
The anwers previously given here (altough correct) I think forget something (may not be a very big thing but this contributed a lot to the general knowledge of Latin).
Latin was still being used in all Catholic sermons until The Second Vatican Council (1962-1965). Until then it was fobidden by the vatican to use the local language in church. Because of this a lot more people knew , altough basic (curch)Latin then there where those who spoke Greek.