We have all been here more often than we would like to be. There comes a point in life when we need to separate. It could be a thing, a person, a role, or any of the things that have been part of our life for a while, and has been of an apparent source of happiness.
The thing that has been providing happiness to us is the last thing we want to part with. That is because the only thing that we all seek is happiness. When the thing stops being a source of happiness, or is not providing the amount of happiness you were used to, then the thing becomes less of a concern. Letting go is not much of an issue. One may miss the moments of happiness that the thing provided in the past, but since the present is no longer the same, we are in a way Ok to let go of it.
Then there are things, that have provided happiness in the past, and seem to have the potential to give you happiness, but that don't seem to. Here there is a struggle. You try to somehow make this thing give you happiness. You do several things, you wait, you hope. Once there comes a point where its no longer giving you happiness, you let go. Since for a while the mind is used to it not giving you happiness , it doesn't bother you much. One may think of this either as letting go, or giving up.
Giving up produces a negative image of the situation. Its like you stopped trying. You keep trying to fix something, or get to some point, and then you stopped it. But if we allow the mind to run its course, it will eventually stop trying and also be comfortable at it. But sometimes a little regret lingers around.
If one thinks about it, it will become clear that Letting Go and Giving Up are essentially the same thing. While the former lends a more graceful transition. The real difference is one strengthens the Ego, while the other weakens it. Letting Go gracefully strengthens the ego, it makes the ego look bigger in spite of the loss. while giving up weakens it. so it is generally easy to assume the position of Letting Go, rather than Giving Up.
Fact of the matter is, separation happens, not because you Let Go, or you Gave Up. It is part of the Change we call life. Change is the secret of this Life. Separation and loss are a mere part of it providing you yet another experience.
Once this is known, and we realize that we neither Let Go of anything, nor do we Give up, but separation 'happens' as part of this long chain of events called life, we begin to enjoy life.