A great English dramatist, Shakespeare, wrote these sonnets.

Both are correctly punctuated. The first says that a great English dramatist wrote these sonnets and the two parenthetical commas enclose the additional information that he was called Shakespeare. The second contains a clause stating that there is (or was) a great English dramatist, then a comma sets that off from the information that this was Shakespeare and he wrote the sonnets.

Compare:

A pink flower, the rose, is loved by many people.
A pink flower, the rose is loved by many people.

Thanks, contrex, for the detailed reply.