Pardon = (formal) forgive; the act of excusing a mistake: â€˜Iâ€™m sure they will pardon the occasional mistake.â€™
Excuse = give someone permission to stay away from school, work etc., or leave a classroom, meeting etc.: â€˜Can I be excused from swimming today please?
I beg your pardon is used (1) to apologize to a stranger because you have bumped into them by accident, sat in their seat by mistake, etc.: â€˜I beg your pardon. I didnâ€™t know the table was reserved.â€™ (2) to politely ask someone to repeat what they have said: â€˜Does this bus go to Marble Arch?â€™ â€˜I beg your pardon?â€™ (3) when someone has said something that makes you feel surprised, shocked, angry etc.: â€˜Whoâ€™s that woman with long hair?â€™ â€˜I beg your pardon! That happens to be my wife.â€™
In formal situations, you can use excuse/forgive me instead of Iâ€™m sorry: â€˜Please excuse me for taking so long to answer your letter.â€™