Giving up and letting go have to do with one's attitude. Some have a tendency to give up easier than others. Giving up is something you do when you experience one or more hindrances on the way, and you decided to quit. Quitting, there is referred to as giving up. For better understanding, here are more synonymous words to what giving up can mean.
Giving up can mean quitting, to surrender, to lose hope, to stop, to desist, to relinquish, to abandon, to capitulate or to admit defeat. Conversely, letting go simply means to release or set free. You can let go by freeing someone, something, an emotion, or a feeling. Letting go is just a perfect antonym to holding on.
That moment you refuse to hold on to something anymore, you are letting go. Letting go is more of a positive attitude than giving up. Giving up can also be a positive act by giving up one's wealth or any other thing for good and peace.