Shakespeare no doubt respected Sophocles' role as the father of tragedy. Just as Sophocles limited the role of the chorus and focused more on humans than gods, so too did Shakespeare reveal secular humanism in his plays.
Shakespeare limits the Chorus to just one, and even he was a minor character, a Prologue Chorus at that. Gone were the days of the Chorus and its leader ever-present on the highest platform of the stage. Shakespeare reserves the "heavens," the highest point on his stage, for the lofty language of love, as evident in the balcony scene in Romeo and Juliet.
Shakespeare breaks from Sophocles for the most part: he uses villains and revenge as the focus of his tragedies. Shakespeare blows up what Aristotle called the three unities: the unity of time, place, and plot. Bill uses multiple days and places to stage his plays. Except in Othello, Shakespeare uses subplots, which Sophocles refused to. Shakespeare's two greatest tragedies, Hamlet and King Lear, are fraught with foils, doubles, mirror images, and dual story-lines. Sophocles' Oedipus is much more straightforward and linear in its tragic fall.
Shakespeare does away with masks, which hinder the speaking of the lines. More, Shakespeare places an emphasis on the words, more than Sophocles does. Shakespeare's plays are long, dramatic poems, not utilitarian or moralistic.
In terms of archetypal characters, they both rely heavily on the Loner (outcast), Temptress, Earth Mother (godess). Shakespeare uses the Villain much more, as well as the Spirit/Intellect. Both Sophocles and Shakespeare rely on numbers, geometric shapes, water, gardens, colors, and celestial bodies (Shakespeare more so). Also, both follow the "quest," "renewal of life," "initiation," "fall," and "sacrifice" for plotting their tragedies. Both rely heavily on eyes (sight, blindness) as the overriding moral motif.