"Screw it !" in politeness/rudeness sits between "Stuff it !" and "Fuck it!"
It is still mild swearing so you need to take care when using it.
"Screw it!" is saying "I am tired of doing it or hearing about it. I am done and finished with it !!"
You've been trying for some time to solve some problem - difficult homework, attempting to build a cupboard that was in kit form, etc, etc, - but you are getting no closer to a solution or getting things to work. You are tired of it all as you are making no progress. You are sick of it and want to give up or move on. "Screw it !!, I'm finished. I'm tired of it all and I'm going to do something else".
To 'screw something up" is a little different in meaning. It means to make a stupid or careless mistake and this mistake has had some consequences. More politely, you would say "stuff something up", more rudely you would say "f. something up".
"I had an argument with my girlfriend so I went back and tried to apologize. But I made things much worse. I really screwed up. Now she's never going to talk to me"
"I was supposed to fix the car, but now it's worse. I really screwed it up as now it won't even start."