When I think of failure in regards to training, I think of training to the point where my form/technique begins to break due to muscle fatigue. Exhaustion is the point where the muscles start to fatigue due to the weight/reps that I am performing. To a degree exhaustion and muscle fatigue go hand and hand depending on your training; if you’re training to failure you will experience exhaustion but might fight through it until your form breaks which will then be the failure stage.
/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.