A Closed Couplet is any complete couplet in which meter and syntax are sealed at the end. The frame is end-stopped. When the lines are written in iambic pentameter and linked by rhyme it is also a heroic couplet.
Know then thyself, presume not God to scan,
The proper study of Mankind is Man.
— Pope‘s Essay on Man(note: this can also be an example of a complete couplet and a heroic couplet.
The Complete Couplet is a poetic unit of 2 lines that expresses a complete thought within itself. Meter and rhyme are at the poet’s discretion.
What is an epigram: a dwarfish whole,
Its body brevity, and wit it soul.
—Samuel Taylor Coleridge, 1772-1834 (note: Because the meter and syntax are end-stopped this is also a closed couplet. Because this example is written in iambic pentameter and linked with rhyme the couplet is also a heroic couplet.)
The Heroic couplet is a complete poetic thought unit of 2 iambic pentameter lines linked by rhyme. It is also a complete couplet and a closed couplet but it is the meter and linking rhyme that sets it apart as a “heroic couplet”. (note: a complete couplet and a closed couplet are only heroic couplets when they are written in iambic pentameter and linked by rhyme.) Shakespeare popularized the declamatory heroic couplet.
“But if thou live remembered not to be,
Die single and thine image dies with thee.”
— William Shakespeare, Sonnet 2
My thanks to Judi Van Gorder for years of work on this fine PMO resource.