Permission asks if the listener will allow the speaker to do something. Request asks if the listener can do something for the speaker.
Permission:
Can I open the window? It’s hot in here.
meaning: Is it okay with you if I open the window?
Request:
Can you open the window? It’s hot in here. meaning: Will you open the window for me?
Note: When we don’t have permission to do something or someone is refusing our request, he/she usually gives a reason why the action cannot be done. It can be very rude to only say something can’t be done without giving a reason why.
Polite:
A: “Can you open the window, please?”
B: “No, sorry. It’s too cold in here.”
Rude:
A: “Can you open the window, please?”
B: “No.”