In Book XI Aurelius wrote:
After Tragedy was introduced the Old Comedy, which through its instructive frankness and its reminder by actual plainness of language to avoid vanity was not without profit, and this directness Diogenes also adopted with a somewhat similar object. After the Old, observe what the Middle Comedy was like and afterwards with what end the New Comedy was adopted, passing little by little into a love of technique based on imitation. It is recognized that there are profitable sayings of these authors also, but after all what was the object to which the whole aim of such poetry and drama looked?
In the passage he mentions Old, Middle, and New Comedies. Do we know whose work he is referring to? And is his criticism of newer works fair?
Aurelius is referring to genre, not specific works, using a - by that point - traditional classification of comedies by time period which is still used today. Reputedly coined by Byzantine scholars, the division works something like this:
Old Comedy - 5th century BC works; our only extant author is Aristophanes but there were many more; traditionally characterised as being political/satirical, with frequently outlandish or fantastical premises.
Middle Comedy - no full texts survive unless we take some of Plautus's Roman comedies as being faithful adaptations of Greek ones; a transitionary period throughout the 4th century BC, about which there is plenty of conjecture as to the form they took. Arnott's chapter on Middle Comedy in the Brill Companion to the Study of Greek Comedy gives a thorough summary of these potential developments, including the de-emphasis of the role of the dramatic chorus and a fondness for piss-takes of mythology.
New Comedy - works from around the time of Menander onwards, heavily influential on Roman comedy. Grounded in reality, not fantasy; farcical, almost sitcom-esque stories about mistaken identities, clever slaves and stupid masters and so forth.
There are two components to Aurelius's criticism that need consideration:
A Love of Technique Based on Imitation
"Imitation" - or "mimesis" in Aurelius's Greek - is a word with heavy philosophical and dramatic connotations stretching back at least to Plato. Mimesis, in the philosophical tradition, is not simply "acting" - it is the performance of an act without actually reaching the truth of it. New Comedy aims for something that reflects everyday life, rather than the fantastical settings of Old Comedy (or the mythological settings of tragedy, which Aurelius writes about in the preceding section of chapter 6), but it is fraudulent. For him, New Comedy is admired for its artifice - antithetical to his own philosophical concerns.
"Profitable Sayings"
Aurelius is using comedy as an example of something that should be able to teach us something.
Aristophanic Comedy is frequently interpreted as having messages for the citizens of Athens; there are moments when the characters break the fourth wall for long speeches about the author or current goings-on in the city, and prominent politicians and contemporary situations are portrayed as major elements in the storylines. No-one is safe from mockery; Aurelius treats this as a reminder to us "to avoid vanity", in your translation above.
New Comedy is "platform agnostic", as it were; there are no real people on stage, just archetypes, and the plots are divorced from contemporary reality. Authors such as Menander, or his Roman successors, might have the odd pithy little turn of phrase i.e. "profitable saying", but there is no outright "instructive frankness" in these plays.
Fair Criticism?
This isn't something that really falls under the banner of "history", in my opinion, but more personal artistic appreciation. Old and New Comedy are very different beasts; a modern reader may rejoice in the ridiculous door-slamming antics of Menaechmi but despair at the pop-culture jokes about Cephisophon's vinegar jars in Frogs because they haven't got a clue what's going on. New Comedy may not be instructional in the same way as Old - but comedy does not always need to have a message to be funny. Aurelius is trying to comment on what art you should like because of the message it gives you (rather than indulging in simple false pleasure) - but that's a decision you have to make for yourself.