These two words are synonyms but they can have different uses. "Erase" is a commonly and widely used term for removing something, normally a mistake. We can erase a written mistake with a pencil eraser, for example. "Expunge" can be the more significant, or serious, removal of something unpleasant, and commonly used in the legal field. For example, if someone's criminal record is expunged, there is no evidence that you were ever convicted of any criminal offense.