/agree with top.
To failure usually implies the amount of reps in a ladder form. Doing as many as possible with a given weight, then lowering the weight and pumping out more, and continuing until there is no juice left.
Exhaustion is different because it has to be reached through a higher weight or with an advanced form of rep.
As an example, HRT is done with exhaustion- specifically in the eccentric phase of the lift.
Pushing the weight in concentric phase then lowering in the eccentric phase for a 5 count forces the muscles to activate in a way that the body is not used to and is forced to compensate for.