When you coax someone, you try to convince him gently, with pleasant words and maybe a little flattery. You’ll have to be patient, as you can’t rush someone you’re trying to coax.
When you coax, you have to be nice about it — you can’t threaten or force. You put on a little charm and gently urge, so that the person or thing is happy about being coaxed. You might coax the runaway elephant back into the zoo by patting her gently and talking into her ear. If you wake up with a terrible bedhead, you might coax your hair back into place with the help of a little hair gel and a comb.