They both mean about the same thing but there can also be a slight difference in usage.
Give up.. means to quit, surrender, relinquish
Give in.. means to relent, to yield, capitulate
Basically the same .. but..
one's goals, on a question, on a riddle, in a fight, something that is treasured
to someone's demands, a debate, to temptation (like chocolate),
For the most part, when somene begs or nags or presses strongly in an argument and the other person capitulates.. use "give in".