I've edited the Reddit comments below lightly to emend typos and punctuation.
Dont_Do_Drama answered this on r/askhistorians on Jul 31 2020.
What a fantastic question!! What you've asked requires that we look into the place that theatrical performances held within each society (i.e. Greek versus Roman). Now, it should be noted that theatre was never static in the ancient world and variations abound between locality and social function (i.e. Athens v. Syracuse--as both were Greek but they practiced theatre in different ways), but there are some comparisons we can draw that will help to provide an answer to your question.
First, let's look at ancient Athens since the examples you provide were from the Athenian Golden Age (i.e. the fifth century BCE) and constitute the foundations of the annual Dionysian festivals that marked the beginning of the trading season in March/April. The Athenian Greeks--and to some extension the Attic peninsula, Athenian colonies, and Athenian allies around in the Greek world--practiced theatre as a part of the festival known as the City Dionysia. For simplicity, I won't go into detail about the City Dionysia except to say that it marked a significant point in the Athenian calendar when citizens gathered to celebrate a patron deity and promote civic engagement among the various classes of Athenian society. As part of the City Dionysia, wealthy patrons (often politicians) sponsored the chori and production of plays that made up the main event of the festival. Essentially, the City Dionysia provided an opportunity where each Athenian demos (*δῆμος)--kind of like the suburbs and neighborhoods of modern urban centers--gathered together in a centralized manner such that the city-state was truly represented as a body-politic within the theatre itself. The actors and playwrights were prominent citizens and the themes they presented were meant to draw attention to particular socio-political issues important to their particular agendas and to the polity of Athenian society who had gathered at the theatre. That is why the Theatre of Dionysus, where the plays were performed, was prominently located on the side of the Acropolis and often served as a meeting place for Athenian political bodies. So, to keep this short, theatre was an opportunity for civic and political engagement that was foundational to the operations of the city of Athens.
This was not the case for the Romans. As you correctly identify, the Romans did not hold theatre to be anything more than a source of entertainment and distraction for the populace of Roman society. But it should be noted that Romans did hold similar (and largely borrowed) ideas about the literary nature of drama in regard to what it could teach a student of rhetorical and oratorical efficacy (see St. Augustine's explanation of his education in theatre in City of God). But perhaps the greatest indication of the role theatre played in Roman society was the lack of purpose-built theatre spaces. As you, again, correctly identify, much of Roman theatre was performed as a part of major festivals celebrating physical prowess (such as the Ludi Romani), marking the superiority of Roman people and Roman society. It wasn't until the 1st century CE that purpose-built theatres appeared in Rome itself! The plays of Terence and Plautus were performed on temporary stages erected in the Roman forum as an engaging form of entertainment--which rarely touched upon politically charged topics. The plays of Aristophanes, at least his earliest plays, were highly critical of socio-political goings on within Athenian society. I'm not an expert on all political things Roman, but the plays (i.e. comedies) of Menander, Terence, and Plautus were much more focused on domestic issues and were more broadly applicable to the Roman merchant classes, lacking the biting political commentary that made Aristophanes so successful in Athens a few centuries earlier. In essence, Rome was by no means attempting to replicate the democratic society of the Athenian Golden Age (however much it held it in esteem) and, as a more authoritarian state, was quick to quash dissent or critique or its political leaders or political acts. Therefore, acting was seen to be much more suspect as a profession. Though a play might have been written down, any performance was much more difficult to control. So, as a succinct conclusion, Roman society largely held drama to be an exercise in mere entertainment than in literary or performative meaning that could be utilized to [eventually] critique its leaders and social organization. So, the people who engaged in theatrical practices were not only suspect to the state, but were maligned in the documentary source that have survived from the early Roman periods (and, likewise, from Roman Christianity--but that's another topic).
The Theatrical Cast of Athens
Performance and Culture in Athenian Democracy
Greek and Roman Actors: Aspects of the Ancient Profession
The Hypocritical Self: Actors, Acting, and Identity in Greek and Roman Culture
Why were Roman dramas and actors judged inferior to Greek ones, when the Romans based their dramas on the Greeks'? : classics
Vertex_21 answered on Jul 27 2020.
Hey, Classics Major here who has studied quite a bit about [R]oman/[G]reek plays. If I were to answer [yo]ur question simply, I[']d probably just say that Greeks and Romans were 2 very different groups of people, with very different identities. Greeks, especially Athenians, identified themselves as sophisticated almost above all else. The arts were incredibly important to them and even their comedies held a lot of weight, usually political.
The Romans, on the other hand, identified themselves as a powerhouse, and this is evident in their entertainment. Roman comedy is far more crude (and that’s saying something!) but, most importantly, it doesn[']t deal with much at all outside of making people laugh. Aristophanes was making comedies such as Lysistrata to try to end wars and make a statement. The Romans were making comedies to impress crowds who may be interested in other activities like the aforementioned lion fights.
This leads into the position of actors in society. If [you're paid to put on a show for a high religious festival where [yo]u will attempt to make a statement about the current political situation, [you're] probably seen in a higher light than someone who acts in order to make a quick buck, think of how we view circus [performers] today. On top of that, most roman actors and writers were slaves. So that plunges them even further down in society.
wooverk answered on Jul 28 2020.
edit: for a great modern example of ideas about the self informing how, generally, a profession is understood, look no further than 'influencing.'
Hey! Others have commented on the political and cultural reasons for Roman distaste of acting but there is also a philosophical reasoning behind the attitude.
The theme of genuity plays right into why actors were looked down upon in Roman society, and particularly looked down upon by an elite class of theatre-goers and historical-record-leavers. The profession was seen as compromising one's own individuality for that of a role; that is, you can't be your true self when much of your time is spent pretending to be someone else. These are the Roman values of being true to one's self, knowing one's self, and being a slave to no one but one's self. In Rome, especially Rome of the mid 1st century CE under Claudius and Nero, stoicism thrived in the imperial court. Though definitely not the origin of disliking actors, mid century stoicism created a philosophical context in which it was popular to expound these types of (anti-acting) values-- being true to one's self, knowing one's self, being one's own master, etc.; see Seneca's letters.
Getting a little beyond your question, it is also interesting that that philosophical context existed right alongside the emperorship of Nero, who was known for his taste for the arts, and his patronage of and performance in public theatre and spectacle. It is likely no coincidence that critiques against Nero and his tendency towards acting and performance are to be found in the works and historical representation of some of the most eminent stoic authors of that time, Seneca, Lucan, and Petronius. Seneca wrote explicitly about these values of selfhood and is historically represented as a curbing force to the fond-of-acting side of Nero's personality. Lucan and Petronius embed their stoicism in their narratives, (Petronius has especially interesting representation of actors/performers), but the idea is the same: The popular philosophy at the time equipped authors, court authors who were also advisors to the emperor, to critique acting and, at the same time, the despot-actor-Emperor Nero. Later historians, like Tacitus and Cassius Dio, really put the nail in the coffin when it comes to accentuating these charactistics of Nero.