I suppose one way of beginning to answer this question is to focus on the emotion that there is in drama as opposed to other forms of literature. There is something about seeing characters in front of you and how they respond to tragedies that allows us to connect with drama in a way that we are not necessarily able to do with prose or poetry. Aristotle refered to this as "catharsis," which he said was a purifying of the emotions through watching drama (in his case tragedy.)
With reference to this excellent play, there is something about watching the heartbroken Maurya acknowledge the death of all of her sons thanks to the sea at the end that resonates with our sensibilities. We see her left alone, insecure and solitary, broken by grief, and yet also we see that in a sense now she is freed from the sea's dominion over her life. Note what she says:
They're all gone now, and there isn't anything more the sea can do to me... I'll have no call now to be up crying and praying when the wind breaks from the south, and you can hear the surf is in the east, and the surf is in the west, making a great stir with the two noises, and they hitting one on the other.
Paradoxically, Maurya, in her grief at the death of her final son and her discovery about the death of her other son at sea, is liberated. Her words thus speak of an immense, age-old sadness of a woman who has lost her child, but at the same time of the way in which the terror that the sea had over her is now no longer binding. The emotional intensity of such lines is what makes drama unique and distinct as we see enacted before us the experience of bereavement.
Regarding drama, one of the key elements I like to focus on is the notion of catharsis. The playwrite Arthur Miller wrote an excellent article on catharsis and its power in drama. Miller's ideas come from Aristotle.
For a work to experience/have catharsis, conflict must arise, tragedy must occur, but hope and acceptance MUST be found. If hope and acceptance are not found, the work has pathos and nothing more.
_Riders to the Sea_ is a powerful play that focuses on the issue of catharsis. Maurya is the prime example of catharis in Synge's play. Notice how the reader first meets her through the words of the daughters; we learn that she is sleeping "if she is able"---remember, she's lost just about her whole family to the sea and she fears she's just lost Michael.
Despite her woes, she does not wallow in her self-pity: her acceptance of fate is never more evident than in her last speech when she says "No man at all can be living forever, and we must be satisfied."
Through Maurya the playwrite Synge gives his reader a catharsis and proves that life goes on.