You're right that these two phrases could be interchangeable in this situation. The meaning isn't exactly the same, though. 'Give me a break' is a fairly literal idiom, and has the sense of 'Leave me alone', or 'Quit bugging me'.
'Cut me some slack' is more idiomatic, and more precise. 'Slack' is the opposite of 'tight'. The image is one of a rope. If you imagine two climbers holding on to a rope, if the climber who has control of the rope cuts the other one some slack they they are loosening their hold and giving the other person more rope to use. When we use the phrase idiomatically, it means to be less strict with someone. For example, if an employee is having problems at home, his/her boss might 'cut them some slack' by being less demanding on them than usual. In your example, the boy is asking his mum to 'back off' a bit (another idiom there) and make allowances for the fact that he's under stress with his schoolwork.