Both Uses

vengeance
in
All the King's Men
(Auto-generated)

A Governor can succeed himself in our state, and the Boss succeeded himself with a vengeance†
Chpt 3 *

with a vengeance = with intensity
Then I was traveling through the part where the crook-legged, high-heeled, gun-wearing, spick-killing, callous-rumped sons of the range live and crowd the drugstore on Saturday night and then all go round the corner to see episode three of "Vengeance on Vinegar Creek," starring Gene Autry as Borax Pete.†
Chpt 7 *

vengeance = the act of taking revenge

Definitions:

(1) (vengeance as in: vengeance is mine)
the act of taking revenge

(Revenge means to harm someone to get them back for something harmful that they have done.)
(2) (with a vengeance as in: with a vengeance)
with intensity
(3) (meaning too rare to warrant focus)