Modern Greek is regarded as the same language as Classical and New Testament times, and in fact in the katharevousa form could be understood by them (albeit Socrates or Paul would probably think you have a weird accent). However, both vocabulary and syntax, and various phonemes, have changed in the 2,000-2,400 years since their time.
Save those who have studied it at the college level, nobody on this board would be able to understand a word of Old English, even though “it’s the same language,” and any randomly chosen post will have more words derived from Latin and the Romance languages than from Old English and the Germanic languages.
Likewise, the evolution of Latin into the Romance languages was a gradual thing. Buying bread and olive oil at the time of Caesar or Cicero, one would not have used the formal language of their writings, but a vulgar Latin of which we have only samples (e.g., caballus rather than equus for "horse.) By the time of Jerome, vulgar Latin was sufficiently the standard that his translation of the Bible was called the Vulgate, because, like modern English translations, it was rendered into the language of the people of the time. A few centuries later than that, with the breakup of the Western Roman Empire, the regional dialects of Latin had imported varying amounts of Germanic-invader vocabulary and moved to the point of bare mutual intelligibility. We date French from the Treaty of Verdun in 842 A.D., the first written use of Old French in formal usage. Italian becomes a language with its own standard with the writings of Petrarch and Dante in the early years of the second millennium. Spanish is not standardized until nearly the time of Columbus, and to a very real extent is still not in a standard form, owing to dialects of near-separate-language-level differentiation. Portuguese has a similar history. Rhaetian still has no standard form, being dialects spoken in parts of Switzerland, Austria, Italy, and Slovenia. Romanian went an entirely separate route, being heavily influenced by its Slavic (and to a lesser extent by its Magyar) neighbors.
But there was not a point when somebody said, “We’re going to stop speaking Latin and start speaking [insert modern Romance language here].” Rather, the common speech moved slowly from Hic haec hoc to L’etat, c’est moi, O sole mio, and Me gusta las chicas.