Good question, Mocchan. Forego (or 'forgo') = not partake of, refrain from "I will forego the pleasantries." "I will forego dessert this evening." Give up (in the sense you mean it) = release possession of something that was had or had been "He gave up smoking." "His father gave up the ghost." (passed away) Forego for when something is possible but won't be taken up. Give-up for when something was but now is not. Hope that makes sense.

Well, forgo is just a synonym with a slight difference. Forgo meant to abstain from or neglect / or to lose , lose the rights to / or to skip Give up is to relinquish your rights to something or to quit. "I will forgo answering your question and allow an example to suffice." "I give up on trying to explain this topic."