What's the difference between rhymer and versifier?

Rhymer

Definition:

(n.) One who makes rhymes; a versifier; -- generally in contempt; a poor poet; a poetaster.

Example Sentences:

(1) I could go on – I haven't mentioned the "blasphemous" Aussie show Come Heckle Christ or Michael Gove's favourite novelty act Mr B the Gentleman Rhymer , back-from-the-dead veterans Jack Dee, Kevin Day and Andy de la Tour, sketch groups such as The Pin, Sheeps and Beasts, new formats such as Will Mars's schtick-swap night Joke Thieves, or this year's candidate for one-off, it-couldn't-happen-anywhere-else special event, Mark Watson's Comedywealth Games – all of which I'm hoping to catch.

Definition:

(n.) One who versifies, or makes verses; as, not every versifier is a poet.
(n.) One who converts into verse; one who expresses in verse the ideas of another written in prose; as, Dr. Watts was a versifier of the Psalms.

Example Sentences:

(1) John Byrom, a lazy, self-indulgent 18th-century versifier, had three black hedgehogs on his coat of arms.
(2) It meant both that Jews might misread poems as anti-semitic and that anti-semites might versify their prejudices.