"Maxim is a general statement, giving some counsel and advice for something useful in life." (Nicolaus 25, Kennedy).Maxim is also a figure of speech and is defined by the author of the Rhetorica ad Herennium as,
"A saying drawn from life which shows concisely either what happens or ought to happen in life." (Ps-Cicero, Rhet. Her. 4.17.24).Maxims are therefore usually short sayings giving some advice. They differ from chreia in two ways: 1) they are always sayings whereas chreias can be either sayings or actions, and 2) Maxims are usually anonymous whereas chreias are always attributed to a specific person.
There is obviously a significant amount of overlap between chreias and maxims, and the distinctions do not seem to be overly important except for classification purposes. This is especially true because all of the exercises used for chreias are also suggested for maxims. Thus, with a maxim, one can expand or contract, elaborate, confirm, refute, and repeat with slightly different language.
If we hold to the strict definitions given by the progymnasmatists, all of the sayings of Jesus are chreias and not strictly speaking maxims. Yet, if one is going by the lists of figures of speech, maxim seems to be the proper figure for the sayings of Jesus.
Several good examples of maxims come in the Sermon on the Mount with the antitheses of Jesus where he says, "You have heard it said... but I say to you." In each case Jesus gives a saying that was common, usually from the Old Testament, but he does not give a specific attribution. In one case, the text is not from the OT:
Matt. 5:43 “You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’In these cases Jesus cites some commonly held saying in the form of a maxim only to refute it with his own chreia.