What is the difference between wheedle, cajole and coax?

They are largely synonymous. Evidenced by the dictionary entries of each of them. ODO lists the other two always in the top three synonyms for all three. The dictionary entries from ODO also hold some usage distinctions.

[NO OBJECT] : wheedle

[WITH OBJECT] : wheedle, coax, cajole

[WITH DIRECT SPEECH] : wheedle, coax

Otherwise you can also look at the further synonyms to derive what other people might derive from the word.

Here's what Merriam-Webster's Dictionary of Synonyms (1984) thinks the differences are:

coax, cajole, wheedle, blandish mean to use ingratiating art in persuading or attempting to persuade. Coax implies gentle, persistent efforts to induce another or to draw what is desired out of another {in a coaxing voice, suited to a nurse soothing a baby—Burney} It most often suggests artful pleading or teasing in an attempt to gain one's ends {little by little, he coaxed some of the men whom the measure concerned most intimately to give in their views—Kipling} {one ... who can linger over and taste a phrase, coaxing its flavor to the palate as if it were an old wine—Moody} {his skill in coaxing ... the attention of the variable human mind to divine objects—T. S. Eliot} but it may be extended to other situations in which persevering yet careful efforts are used to attain an end {coax embers into a blaze} Cajole may stress deceit (as by flattering or making specious promises) {they ... should be treated as they themselves treat fools, this is, be cajoled with praises—Pope} It more often implies enticing or alluring and suggests beguilement rather than duplicity {I think a vein of sentiment ... induced me to take the journey, and to cajole a reluctant friend into joining me—Repplier} Wheedle suggests more strongly than cajole the use of soft words, artful flattery, or seductive appeal {she could wheedle the soul out of a saint—Hewlett} {he had wheedled the Abeyta woman out of her geraniums, and left her pleased with herself for surrendering them—Mary Austin} {no hucksters to wheedle you into buying souvenirs—Nebel}

So MW chiefly distinguishes between the three words in terms of gentle persistence in artful pleading (coax) versus enticement through beguiling maneuvers or duplicity (cajole) versus artful or seductive flattery (wheedle). Persistence seems to be central to all three forms of persuasion, and I suspect that there is a great deal of overlap in how people use these terms in everyday speech.