The allusions. Which are much more obvious in Ancient Greek, because it had several quite distinct literary dialects. If you want to allude to Homer, or to the tragedians, you can easily choose a word that occurs only in Homer, or a grammatical inflection that is antiquated. And literate Ancient Greeks were meant to be across all the canonical texts; so one adjective can invoke an entire myth. (It’s no different in our contemporary cultures; we just have different canons.)
The convoluted syntactic structures: how, at its best, a prose sentence is a poised, beautiful construct, with lots of nesting and embedding and qualifications and rhetorical contrasts. And at its worst, it’s a rambling, ugly jumble, with lots of nesting and embedding and qualifications and rhetorical contrasts. We don’t write like that any more, and more and more, we don’t read like that any more. Not necessarily a bad thing, just different.
The subtlety of free word order; something I miss in translation from Modern Greek as well. Free word order isn’t just an excuse to put syntax in the blender (at least, it isn’t in prose); it’s a way of making nice, understated distinctions in emphasis, or contrast, or in topic vs comment structures in the sentence. English does that as well, of course, but with different means: there’s a reason plaintext English still feels it needs to use all caps or asterisks for emphasis, and Greek doesn’t.
The kind of primitive way Classical Greek deals with abstractions: it’s not Goodness, it’s The Good; not Equality, but The Equal. In fact, old technical Greek seems to make do with some surprisingly sparse resources.