Both of these words mean to ask strongly, with a lot of deep emotion:
The word beg (but not plead) is also used when poor people ask for money on the street: they are begging, and someone who does this frequently is a beggar.
The word plead (but not beg) is used in the judicial system, when someone who is accused of a crime pleads innocent or pleads guilty (officially states that they are innocent or guilty).
We have two idioms with beg:
However, in everyday spoken English it’s more common to use “Excuse me” instead of “I beg your pardon.”