Is there a big difference between Modern Greek and Medieval Greek?

Non-zero, but not huge.

Mediaeval Greek is  not the normal term used, because the Greek linguistic situation doesn’t align well with the Middle Ages. Let me explain.

The learnèd language of Byzantium was Attic Greek, with varying degrees of enthusiastic hypercorrection and exoticism. The officialese language of Byzantium was closer to Koine, with plenty of Latin terms.

The vernacular language is not well attested. We have the papyri up until around 700, a couple of generations after the Arab conquest of Egypt—although they represent one region and what was often a second language. Our first vernacular texts are from the mid 12th century, but all vernacular literature is macaronic with more archaic Greek, up until at least the Cretan renaissance.

Betweem 700 and 1150, we have some acclamations and ditties from chronicles, and the Proto-Bulgarian inscriptions, likely written by Greek POWs.

Received wisdom is: the Greek vernacular switched from Middle Greek to Modern Greek  in the Dark Ages—so around 1000.

The pronunciation of Greek, at a phonemic level, was pretty close to Modern by 500; the final change, of οι and υ to /i/, seems to have been around 1000 (a poem makes fun of the new pronunciation in 1040; but these things always take longer—place name evidence shows the old pronunciation lingering for another century). I’m not sure we know when double consonants were degeminated—and of course they haven’t degeminated in all Greek dialects.

The morphology seems mostly modern by the start of Early Modern Greek. The dative and future tense is still to be seen in the bits between 700 and 1150, but is gone by the first Early Modern texts. Verb tenses took a while to settle down: the future particle θα is 17th century, and the volitive future it came from (θέλει Verb) is 14th century; the perfect is 17th century, the pluperfect even later; the conditional used to use the aorist instead of the imperfect (να είδες, not να έβλεπες). The positioning of clitics was not the modern positioning; as with other features of Early Modern Greek, it was closer to Modern Cypriot than to Standard Modern Greek.

So the differences between 1200 Greek and 2000 Greek are noticeable. But I have to say, they’re not huge. The vocab is a bit on the exotic side; but within the range of modern dialect. If I were to compare it to English, maybe Shakespearean English, maybe later. (The tense “was being built” after all is only 19th century in English.) If you have some linguistic smarts, you’ll understand it just fine.

Did Hebrew influence Ancient Greek?

Thx4A2A, Dimitra.

The mainstream of Greek was not influenced substantially by Hebrew. The Hebrews were just another barbaric tribe in Classical times, as far as the Greeks were concerned, and not a terribly important one. Greek did get some words from Persian (the word for “chore”, αγγαρεία, is still used); but the Persians had an empire: they were barbarians that the Greeks noticed.

Because of Christianity, Koine Greek did receive some Hebrew lexicon, but really, not that much: most New Testament writers made a point of translating things into Greek. The most enduring influence on Greek is probably the word for Saturday, Σάββατο, which is of course just Sabbath; and a swag of ultimately Hebrew names, like John and James.

The Greek that Jews spoke was of course a different story. The Greek of the Septuagint is at times awkward, in some part because it’s a literalish translation, in part because of Semitisms in the language. Some of the New Testament is quite learned Greek; but some is colloquial vernacular, and again there are expressions showing Hebrew and Aramaic influence. (More Aramaic.)

We don’t know enough about Judaeo-Greek, the now extinct Greek spoken by Romaniote Jews in modern times. The Judaeo-Greek Torah of 1547 was a word-for-word translation; being in vowel-pointed Hebrew script, it tells us a fair bit about Early Modern Greek phonetics, but nothing about any living language’s syntax.

But from what we do know, the relation of Judaeo-Greek to Christian Greek was not like Yiddish to German, and indeed few Jewish languages were as distant from their Christian counterparts: they were more like Jewish English, with some lexical loans from Hebrew, and maybe some archaisms from being separated, but otherwise identifiably the same language. As this article shows, codeswitching to Hebrew came in handy for secrecy—as indeed Greek migrants have done elsewhere.