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.
There is a big difference that you should know about giving up and letting go. Giving up means that you are stopping the process which you feel will allow you to reach a new path. This means that you are allowing your struggles to just stop you from pursuing what you want because you feel that it is not something that is meant for you anymore. Letting go means that you are allowing yourself to be free from something that is not providing you with what you need anymore.
For example, you feel that your relationship with a friend is not serving you well anymore. This means that you are letting go of that friend because you are not getting anything from that friend anymore.