What are the common elements of Elizabethan tragedy versus Shakespearean tragedy?

Although this question is under the subject of Shakespeare's sonnets, the question itself seems to focus exclusively on drama. However, I will try to include an analysis of the sonnets also, insofar as they relate to the plays of Shakespeare and his contemporaries.
First, though Shakespeare himself was part of the Elizabethan period, some of his plays, such as Macbeth, The Tempest, and Henry VIII, were written after Elizabeth's death, when James I was on the throne. Your question, insofar as it relates to drama, can thus be divided into two parts. First, during the actual Elizabethan period, what similarities and differences could be seen between Shakespeare's plays and those of his contemporaries in London theater? Second, what common elements and differences exist between Elizabethan plays, by others or by Shakespeare himself, and the plays written after 1603 during James's reign (the Jacobean period)?
The answer to the first question, as metrical form goes, is that Shakespeare, like his contemporaries Thomas Kyd, Christopher Marlowe, and others, writes chiefly in blank verse (unrhymed iambic pentameter), although he does have prose interpolations as well as occasional rhymed passages. In their tragedies, all of these playwrights deal with similar themes: history, both recent and ancient, and violent subjects often involving revenge and murder. In his earliest plays, such as Titus Andronicus, Shakespeare does not reveal much to distinguish himself from the other playwrights. Shakespeare's language at this point (in the early 1590s) does not yet show the complexity of the later pieces, and it is simpler and more direct, though it is of course highly poetic, like that of Marlowe. This is true of Romeo and Juliet, for example.
In the plays written well after Marlowe's and Kyd's deaths, Shakespeare begins to use words in a more striking and imaginative way and to develop the kind of idiosyncratic language that is one of his chief assets, raising him above other dramatists and poets. Hamlet and Othello, among others, date from around the turn of the century during Queen Elizabeth's last years. The greatest of his plays dating from the Jacobean period is probably Macbeth. We can distinguish Macbeth from his three other greatest tragedies (Hamlet, Othello, and King Lear) because it is shorter, less discursive, and has a more streamlined plot.
One also can see that Shakespeare was not above trying to curry favor with the ruling house. During Elizabeth's rule, he wrote Richard III, which fictionalizes many elements of the Wars of the Roses in order to make the Yorkist party, defeated by the Queen's grandfather, Henry VII, look especially bad. In Macbeth, we see a presentation of Banquo, an apparent ancestor of King James, as a hero/victim of Macbeth's lust for power.
In spite of changes in subject matter and language, the later plays are still written in blank verse and deal with the same general kinds of subjects that had been popular on the English stage for a long time. All of them use preexisting material: for the English history subjects, the chronicles of Raphael Holinshed are used; for plays taking place in Italy, romances by Italian writers; for Hamlet, the Danish history by Saxo Grammaticus; for ancient history, chiefly the Lives of Plutarch. At that time, there was no stigma attached to drawing on previous authors' works for one's material.
Several things can be considered in looking at Shakespeare's sonnets in relation to his tragedies. Each sonnet can be viewed as a miniature drama of its own with a specific theme or a "point" to make. For instance, Sonnet 33 begins as follows:

Full many a glorious morning have I seen
Flatter the mountaintops with sovereign eye,

However, this seems to make the point that all of this is an illusion and that such splendor does not last. Sonnet 129 can be viewed, in some sense, as a fourteen-line analog to Hamlet:

Th' expense of spirit in a waste of shame
Is lust in action; and till action, lust
Is perjur'd, murd'rous, bloody, full of blame.

However, the "action" alluded to may not be of the same kind as Hamlet's planned vengeance against Claudius. A particularly interesting sonnet is Sonnet 130, "My mistress's eyes are nothing like the sun," for it gives us a point of comparison with another great writer of sonnets of Elizabeth's reign, Edmund Spenser. Shakespeare, arguably, is debunking the courtly, idealized view of women his predecessors expressed, though he is doing this not cynically, but, rather, in praise of a woman. Spenser, as beautiful as his sonnets in Amoretti are, makes his love a kind of mythical goddess. Within a few short years after Spenser, poets began to adopt a much more realistic attitude toward love.
A last point about the sonnets is that while each one can be considered a small drama in itself, the body of sonnets as a whole has a kind of loose story, and it contains "characters," whom scholars have referred to as the Dark Lady, the Fair Youth, and The Rival Poet. It is difficult to make out a coherent story of it, however, but perhaps for that very reason, the sonnets, in their mysterious, dreamlike way, are all the more fascinating.
To sum up, Shakespeare's own tragedies evolved and became more complex in language and overall artistry between the earliest ones and those like Macbeth, written later. It would be too much of a generalization to assert that there are specific characteristics of the post-Elizabethan plays as a whole, except for topical references and flattery directed to the King personally. In subject matter and basic technique, Shakespeare is clearly of his time and is similar to his contemporaries. But Shakespeare's language is richer, more complex, and more unusual than that of the other tragedians, even Marlowe. Finally, the sonnets have nearly the same richness of language as Shakespeare's tragedies, and they can be seen as analogs in miniature to the plays.