The main difference between giving up & letting go is the energy & intention behind it.
Many people think that giving up & letting go are the same thing but they are not.
Giving up is often a result of frustration & anger. It is a forced capitulation. Sometimes an act of defiance. It can also be the moment a resentment is born. It starts to breed negativity.
"I can't make this work, I give up"
"You won, I lost. I give up"
"I failed, I give up"
We say to ourselves we have "given up" but secretly we hold on to the hope that if we stay annoyed enough, bitter enough & assign enough blame outside ourselves then things may magically turn around. But negative thinking almost never produces positive results.
Letting go on the other hand is a trust that everything is going to be OK. A peace within ourselves that we are doing the right thing, because we feel like we not being forced.
Letting go is less like a strong thought & more like a feeling. It's not really something we have to say to ourselves. It just is.
It's really hard to do. It's so much easier to say in our head that we have given up & then try to convince ourselves that we are at peace with that decision.
Yet when we let something go (without the hope it will magically fix itself), it could be a person, project, an idea or belief, we allow better opportunities to enter our lives.
I'm sure that it is no coincidence that the business owners I work closely with are able to let go easily, and move on quickly. They have happier staff & more productive businesses because of it.