To run away, or escape privately, from the place or station to which one is bound by duty; -- said especially of a woman or a man, either married or unmarried, who runs away with a paramour or a sweetheart.
To elope, most literally, merely means to run away. More specifically, elopement is often used to refer to a marriage conducted in sudden and secretive fashion, usually involving hurried flight away from one's place of residence.