Pardon

Pardon = (formal) forgive; the act of excusing a mistake: ‘I’m sure they will pardon the occasional mistake.’

Excuse = give someone permission to stay away from school, work etc., or leave a classroom, meeting etc.: ‘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.: ‘I beg your pardon. I didn’t know the table was reserved.’ (2) to politely ask someone to repeat what they have said: ‘Does this bus go to Marble Arch?’ ‘I beg your pardon?’ (3) when someone has said something that makes you feel surprised, shocked, angry etc.: ‘Who’s that woman with long hair?’ ‘I beg your pardon! That happens to be my wife.’

In formal situations, you can use excuse/forgive me instead of I’m sorry: ‘Please excuse me for taking so long to answer your letter.’