Giving In, you are making a decision to agree with someone, or do something or maybe not do something because you agree to the person making the demand / request. You may want to do this for love, or just to shut the other person up, but still you want to do it is a nice thing.
Giving Up, on the other hand is defeatist. When you give up, you essentially tell yourself that it is not worth it. That feeling will not go away, not ever.