This is not just a fancy word for a pun, whatever certain works of reference or internet misinformants may think.  It is generic for any play on the sounds of words, of which the pun seems a trivial form and can be quite silly. Charles Lamb wrote that a pun ‘is a pistol let off at the ear; not a feather to tickle the intellect.’ (Last Essays of Elia: Popular Fallacies, No.9)

The purpose of a pun is usually just the pun; but paronomasia can be solemn or even sad, when it is used by speakers or writers to make the assertion itself memorable—which is the motive even for ‘Beanz Meanz Heinz’.

A few examples:

  • non Angli, sed Angeli: these are the words commonly attributed to Gregory the Great when he saw some English slave children for sale in Rome. What Bede actually says is that when Gregory asked where the children came from,
    ‘responsum est, quod Angli vocarentur. at ille, “bene,” inquit; “nam et angelicam habent faciem, et tales angelorum in caelis decet esse coheredes.” Bede, Historia Ecclesiastica II.1
  • urbi et orbi: the solemn blessing by the Pope from the balcony of St Peter’s.
  • modo patet, modo latet: source unknown.
    Now you see it*, now you don’t.
  • πς γρ οχὶ γεννάδας,
    στις γε πίνειν οδε κα βινεν μόνον;
    Xanthias, describing his master Dionysus in The Frogs, line 740:
    Yes, of course he’s a gentleman,
    his skills are only drinking and bonking.
  • tibi erunt parata verba, huic homini verbera: said by the slave Syrus to his master Clitipho, warning him that they may ‘catch it’: huic homini refers to himself.
    Terence, Heauton Timoroumenos, line 115.
    Translated by one student as “You’ll get a tongue lashing, I’ll get a good thrashing.”
  • dum spiro, spero: Roman proverb.
  • I love thilke lasse, (alas, why doe I love?): the lament of Colin Cloute in Spenser’s The Shepheardes Calender: Januarye.

*or him, or her.