Aeschylus is the most traditional of the three. He expanded the role of the cast, from one actor to two. His plays still have the Chorus front and center, as protagonists. He is mainly theological.
Sophocles is in the middle of these two. He is half-traditional and half-modern. He has more actors (3-4) than Aeschylus, but not as many as Euripedes. He expands the role of the Chorus, but they are not protagonists--more objective, in the middle. His plays focus on strong women (Antigone). His plays are humanistic and fairly political.
Euripides is the most modern of the three. He limits the role of the chorus the most, focusing on the inner lives of his characters. His characters seem a departure from the classic tragic heroes, as he places strong women (Medea) at the forefront. He even satirizes the Greek heroes of old. According to my notes:
[Euripides'] plays are more exuberant than those of Sophocles and Aeschylus; often, he has the heroes and heroines face difficult choices, which are finally solved by the sudden appearance of a god (deus ex machina).
So, overall, the role of the chorus seems to be the most significant difference between these three. Enotes says it best:
However, the role of the chorus changed over time and in the hands of the three great tragedians. For Aeschylus, the chorus played a more central role. In the Suppliants, the chorus is actually the protagonist, while in Agamemnon, the play’s themes find clearest expression in the vocalizations of the chorus. In Sophoclean drama, the chorus could be interpreted as a group of characters itself, with a distinct perception and point of view. In some of Sophocles’s plays, as in Ajax and Electra, the chorus was most closely attached to the title character. In other plays, namely Antigone and Oedipus at Colonus, the choruses are made up of city elders who present their opinions on the events they are witnessing. By the time of Euripides, however, the chorus had taken on a far less crucial role. According to Rex Warner writing in Three Great Plays of Euripides, in the works of Euripides, “The chorus perform in the role of sympathetic listeners and commentators, or provide the audience with a kind of musical and poetic relief from the difficulties or horrors of the action.”