"Cave in" means that you allowed yourself to influenced by external pressure. i.e. the weight of the mountain caused the crave to crushed. Give up is more normal than give in. "Give up" is to stop trying to do something becasue of any reason. it could be an external source e.g. peer pressure or an internal source e.g. bored howver it typically means the person in question deicides to give up. So the difference between "cave in" vs "give in to someone" basically is that when someone caves in they didn't do it by choice they were overcome by external pressure/factors and could cope. "Give in" implies that they could have kept going but decided not to. At least that's the difference for me.