vengence is a feeling for revenge, revenge is the act.
They are the same, except, vengeance is something you feel and revenge is the plot you employ to get your vengeance.
Vengeance is the act of taking revenge.
They are extremely similar...I would have to say revenge involves injury and/or avenging with equal degree to the offense (Such as somebody steals your bike so you steal theirs). Vengeance is more like punishment in retaliation of an injury and/or offense. Plus I have always considered vengeance to be the extreme of the two...(Such as someone beats you up and you respond by killing them)...