I’ve been (somewhat casually) learning Hellenistic/Koine Greek over the past few years, and to a certain extent, Classical/Attic Greek. I’ve only taken two actual semesters of Greek, but have supplemented that with a decent amount of extra studying on my own. Becoming more and more fascinated by the Greek language, I recently decided to begin learning modern Greek. Not surprisingly, there are some noticeable differences between modern Greek and ancient Greek. Here are a few I have noticed so far.

The Greek alphabet has basically remained the same from Hellenistic Greek onwards, although there have been changes in orthographic conventions and considerable changes in the pronunciation of the letters, for instance [letters in square brackets represent pronunciation using the International Phonetic Alphabet]:

• the letter beta, β,  was [b], but is now [v].
• the letter gamma, γ, was [g], but is now [ɣ] before an α, ο, or υ, and a [ʝ] before an η, ι, υ, οι, and ει [tangent: the popular Greek yoghurt brand, Fage, is actually pronounced as Fa-yeh]. The double gamma, γγ, is a pain and is variously pronounced (depending on what it appears before) as [ŋg], [ŋɣ], [ŋʝ], or [ŋɟ].
• the letter delta, δ, was [d], but is now [ð].
• the letter theta, θ, was [tʰ], but is now [θ].
• the letter phi, φ, was [pʰ], but is now [f].
• the letter chi, χ, was [kʰ], but is now [x] and [ç].

Keep in mind that this is how they’re pronounced in Standard Modern Greek. I’m sure that the actual realization of consonants varies across Greece and Cyprus.

The [b] to [v] change in beta was the first change I noticed. Apparently, though, it is not that uncommon for a plosive consonant [b] to turn into a fricative consonant [v]. Delving into linguistics (especially historical linguistics) has been a very intriguing and rewarding venture, and it is kind of amazing that there are particular tendencies which seem to govern phonological change in many languages over time.

Additional changes are:

• the disappearance of the rough breathing mark [h].
• many vowels and diphthongs (e.g. ι, η, υ, ει, ει, οι, υι, ῃ) have simply turned into [i], while ε and αι have turned into [e], and ου is now [u], and ω has turned into [o]. In modern Greek, ειρηνη (peace) is pronounced something like ειρινι, with the η’s and the ει diphthong now being pronounced simply as ι (whereas they used to have distinct sounds from ι). This shift of the vowels to being pronounced as ι is known as iotacism/itacism.
• an upsilon, υ, is pronounced as [i], though when following another vowel it is pronounced as [v] or [f], whereas it used to be more like the German ü [y]. For example, in the initial syllable in the word for “thanks,” ευχαριστω, the ευ used to be pronounced as it appears in “feud”, but it is now pronounced as “ef-”. Before vowels and voiced consonants (δ, γ, ζ, λ, ρ, μ, and ν) ευ is pronounced as “ev-” (e.g. ευαγγελιον), and before a  β and φ, ευ is pronounced as [e]. In non-diphthongs the solitary υ is pronounced as [i].
• the dative case is no longer in use in modern Greek (its function being absorbed by the genitive).
• the optative mood has all but dropped out of use (except in fixed expressions).
• only the acute accent appears in modern Greek, with the other two accents (grave and circumflex) officially dropping out use a few decades ago in 1982.
• the nominative form of many third-declension nouns has changed.

I’m sure there are many more changes I will come across as I slowly progress in learning modern Greek, but these are some of the more noticeable ones that jumped out at me.