Yes, I know that there is debate about whether or not describing Ancient Greek as ‘dead’ is appropriate or not. I am sympathetic to some of the arguments that Ancient Greek is not ‘dead’. But no remotely-knowledgeable person would seriously question whether or not Mandarin is a ‘living’ language. I’m going to refer to Ancient Greek as ‘dead’ because it is much shorter than ‘totally lacking in living fluent speakers who use the language for everyday purposes, such as new popular fiction, communicating with tax collectors, scolding children, finding the nearest bathroom, and eavesdropping’.
(By ‘Ancient Greek’ I mean everything from the Archaic Dialect of Homer to the Koine Dialect which includes the New Testament – I’m not including Mycenean Greek from before Homer, or Greek that is newer than Koine).
Learning a dead language is easier in the sense that you don’t have to worry about seemingly subtle mistakes sticking out like a sore thumb to native speakers, or even learn any listening or speaking skills. Learning something about Ancient Greek pronunciation is necessary to understand certain points of grammar and to appreciate the poetry, but since nobody nowadays is certain how people in 4th century B.C. Athens pronounced the pitch accents, there is zero pressure to master that kind of thing. By contrast, messing up tones in Mandarin can cause problems. In southern dialects of Mandarin (including the Taiwanese dialect), the only difference between the word for ‘four’ and ‘ten’ is a tone, so I’m in the habit of making hand gestures to indicate whether I mean ‘four’ or ‘ten’ just in case I’m messing up that tone (I indicate ‘four’ by holding up four fingers, I indicate ‘ten’ by crossing my index fingers, which most Chinese speakers understand).
But in the long run, this makes learning dead languages harder, not easier.
When learning living language, you have to resign yourself to embarrassing yourself by speaking in a ‘broken’ way at least for a while, and possibly for as long as you ever use that language. It sucks. But it is good for learning. If you have been humiliated by a particular mistake, you’re likely to remember it, and maybe not repeat the mistake. I know that I vividly remember some situations I’ve gotten into by speaking bad Mandarin. And you’re motivated to learn to prevent humiliation. There’s some feedback and accountability (though it’s not reliable).
With dead languages, that doesn’t happen. Sure, a teacher can spot and point out mistakes, and that’s helpful for learning the language, but it’s not the same thing as trying to sell books to a used bookstore, and the worker points you to the shelves because they think you want to buy books (that didn’t actually happen to me, but I can imagine it happening).
Though Modern Greek is not the same as Ancient Greek, the writer of this essay found that learning Modern Greek improved her understanding of Ancient Greek, partially because it’s a closely related language which can be practiced by having everyday banal interactions. And it has this quote:
Classicists, he suggested, are easily embarrassed and afraid to make mistakes. Making mistakes is crucial for language acquisition, and sometimes the mistakes will be horribly embarrassing ones (I have, in polite conversation, said τσιμπούκι when I meant τσιμπούρι). Once, after I paid for books at a bookstore in Greece, I overheard the woman who had just rung me out ask a colleague with genuine bewilderment: “What does she want with an Ancient Greek book if she can’t even speak Greek?” In a field that already demands so much posturing, so much pretense of knowing Greek and Latin, risking mistakes and “not knowing” means risking a lot of your ego.
But the difference between learning a dead language and a living one goes beyond the opportunity to use a language in everyday situations and the likelihood that mistakes will be brought to a learner’s attention. It’s that there are simply more opportunities to use the language in more ways when the language is widely spoken. Someone learning Mandarin can watch movies and TV shows in Mandarin, listen to pop songs and podcasts in Mandarin, play electronic games in Mandarin, casually chat with people face to face, or on the internet, or by phone, in Mandarin, and so forth. None of that is really possible with Ancient Greek. Sure, there are a few people who try to chat with each other in Ancient Greek, but it’s awkward, and there may even be computer games in Ancient Greek, but they are probably very few in number, and may not be much fun to play. And even though there are recordings in Ancient Greek with reconstructed pronunciation, the pickings are slim.
Now that I am returning to studying Ancient Greek after putting a ton of effort into learning Mandarin, I am really missing all of the diverse opportunities to practice Mandarin which don’t practically exist for Ancient Greek. If I were more serious about Ancient Greek, I might even follow up on that essay writer’s suggestion to learn some Modern Greek to improve my ‘feel’ for Ancient Greek.
If I didn’t have the experience of studying both a living and a dead language, I don’t think I would really appreciate the difference, and I’m not sure that I’ve done a good job of explaining to those of you who haven’t had the experience yourselves.
The Notes Which Do Not Fit 
One of the main differences that I noticed when it comes to studying dead languages (4 years of high school Latin) vs. living languages (1 year of middle school Spanish; 2 years of college Japanese) is the amount of emphasis put on like, linguistics and grammar and the underlying theory of language, vs. the ability to use the language in practical situations.
In most of my classes that focused on living language, the goal was basically to get you to the point where you could have basic conversations as fast as possible – so there was a lot of emphasis on rote memorization of vocabulary and common conversational phrases, with only a little bit of syntax, the answer to many “why does it do that” questions was “it doesn’t really matter, just memorize it.” Most living language programs tend to put intensive grammar/linguistics study at the end of advanced courses, rather than explaining these things to beginners. And this makes sense, since actually being able to use the language is often the goal of many people taking such courses, and if you just want to be able to muddle through basic conversations you don’t necessarily need a thorough knowledge of the syntactic underpinnings of the language.
On the other hand, with a dead language, I’ve noticed that there’s much less emphasis on rote memorization of vocabulary and phrases (after all, if you’re just translating old records, you can stop and check a dictionary as often as you need – the document won’t care) and many of the texts were open-book with free access to dictionaries. That leaves much more time for more intensive discussion of things like syntax/grammar, as well as general linguistics principles. So there’s a lot more time (for classical Latin at least) talking about things like declination and conjugation, what things like gerundives and infinitives are, what nominative case is or what vocatives are, what agreement and valence are, etc. This also involves talking about the equivalent constructs in English so that you can translate appropriately back and forth; which is how I learned way more about English grammar and linguistics in Latin class than I ever did in any of my actual English classes.
My experience is a bit different. I remember quite a bit of focus on grammar / conjugation / etc. in my high school French class (and I also learned way more about English grammar/linguistics from studying French than from my actual English classes). There were also a lot of conversational phrases, but not so great a focus on building vocabulary. By contrast, the main textbook I used as a teenager for Ancient Greek, Homeric Greek by Clyde Pharr (which obviously teaches Archaic Greek and not Attic Greek, Koine Greek, etc.) there is a strong emphasis on learning lots of vocabulary, because there simply isn’t any way to read Homer without that. Technically, it’s possible to pull out the lexicon every time you don’t recognize a word or you recognize a word but don’t understand the usage. But without a large vocabulary, you would have to pull out the lexicon so often that it would take forever to read something and you’d lose the train of thought. I used to have this book that was just about the most frequently used words in Homer, it’s that much of an issue for people reading the Iliad and/or Odyssey in the original.
I don’t remember how many unique words are used in the Iliad/Odyssey, but the number which I would guess is 10,000 words. My copy of Cunliffe’s Lexicon of the Homeric Dialect, which only has an entry for a word if it appears in the Iliad and/or Odyssey, is more than four hundred pages long, and it’s printed in small type. Only a dedicated scholar/fan would learn the entire vocabulary, but knowing ~2000 words is considered the bare minimum to be able to read Homer with a good flow.
That said, the textbook which I used recently to restart my Greek, Greek: An Intensive Course, is more like you describe – very strong emphasis on grammar, limited vocabulary. It’s Attic Greek, and geared towards getting students to read writers like Plato and Xenophon (who did not have ridiculously rich vocabularies) as opposed to writers like Aeschylus (who did have a ridiculously rich vocabulary). It’s possible that Latin is less vocabulary-dense than, say, Archaic Greek, so students don’t have to put quite as much effort into vocabulary building to be able to read competently.
I know that some teachers do try to teach Ancient Greek more like a living language – using conversational phrases, dialogues, etc. Since some of the surviving Ancient Greek literature is conversational (particularly Plato and Aristophanes), it can tie into the texts pretty smoothly.