"Give in" is more to acknowledge defeat or yield to an opponent of some sort.
"Give up" is more like to abandon hope and stop trying.
Here are some examples
Give in:
1. Its leaders seem to think this military saber-rattling will force other countries to give in to their demands.
2. I have to keep my mind clear and not give in to emotion.
Give up:
1. Tough years often cause investors to give up on stocks.
2. Part of an agent's job is to never give up, but we need the same from our clients.