I love this question! And I’m on mobile (because my computer is dead and I’m nowhere near a power outlet—mods be merciful) so please bear with me.

I’ve provided bits of an answer to this question in other threads. I’ll link to them later, when I’m able to have full computer POWER. But suffice it to say that the Romans were a significant influence in two ways: domestic comedy and the dramatic shape of tragedy.

Let’s start with the former. The plays of dramatists Terence and Plautus have survived thanks to the diligent copying of medieval scribes, who sought to preserve the light-hearted Latinity for students in monastic and cathedral schools learning conversational oration and performative comportment in their duties as courtly officials across Europe. Perhaps the greatest proponent of this history is C. Stephen Jaeger, especially his seminal work, The Envy of Angels. Nevertheless, we have Terence and Plautus to thank for the stock characters and situational comedy that we are some familiar with today. From overbearing parents to rowdy teenage children to witty and intelligent servants/slaves (i.e. underclasses) trying to correct their obtuse “betters,” it’s all there in the domestically-centered comedies of Latin playwrights. From the Middle Ages to the modern period those TYPES of characters continue to survive and are still a part of our dramatic performances today. Now, I do have to mention that the Romans didn’t invent this comic structure. The Greeks, specially Aristophanes, preceded them. But thanks to the work of Menander and, later, Terence and Plautus, the Romans really perfected this type of comic structure.

Turning to the shape of tragedy: we have Seneca to thank for the 5-act tragic form that became especially popular in the early modern period—particularly with this one playwright you may have heard of named Bill T. Shakespeare (no idea if his middle name began with a T. but that’s what i prefer so I don’t care what the mods think 😬.)So, Seneca (lived 1st century CE) [probably] wrote a number of plays that shot to popularity in the early modern period (16th century CE). In particular, he emphasized the 5-act structure of tragedy and popularized the concept of a central figure driven by revenge to advance the plot of the play (i.e. Hieronimo in The Spanish Tragedy or Hamlet in Hamlet). The whole speech of Liam Neilsen in Taken is like the spotlight monologue from a Senecan tragedy (except—SPOILER ALERT—his daughter lives). Seneca, unlike his fellow Greek playwrights, was not afraid to advocate for violence and gore on the stage (e.g. Hercules furens).

Yeah, so, all in all, Roman dramatists were certainly building upon the dramatic foundations of their Greek forebearers but they provided some uniquely Latin/Roman qualities to playwriting that would influence generations of dramatists up to today.

I’ll come back to edit this with links and more references later. Sorry mods!!!

EDIT: I'm back with references (below)!

Richard F. Hardin, Plautus and the English Renaissance of Comedy (2017).

Evangelos Karakasis, Terence and the Language of Roman Comedy (2005).